mental Physics 7c;LjiKw;; E. SEGRE Editor . EXPERIMENTAL NUCLEAR PHYSICS CONTRIBUTORS VOLUME I Julius Ashkin Kenneth T. Bainbridge Hans A. Bethe Norman F. Ramsey Hans H. Staub VOLUME II Philip Morrison Bernard T. Feld VOLUME III In preparation EXPERIMENTAL NUCLEAR PHYSICS VOLUME II E. SEGRE, Editor P. Morrison and B. T. Feld JOHN WILEY & SONS, INC., NEW YORK CHAPMAN & HALL, LIMITED, LONDON few. i £i>/ CLASSICS b,-^C NO. , ■„JjJ4^Ss _ Checked I ^te*V *f /u-*>-70 Copyright, 1953 BY John Wiley & Sons, Inc. All Rights Reserved This book or any part thereof must not be reproduced in any form without the written permission of the publisher. Library of Congress Catalog Card Number: 52-5852 PRINTED IN THE UNITED STATES OP AMERICA PREFACE At the end of World War II many physicists who had been mobilized for the war effort returned to university work and to pure research; a great number of them had worked on nuclear problems and were anxious to resume investigations in this field. Moreover there was a large influx of students eager to start nuclear investigations. The need was keenly felt for a book which would bring the experimen- talist up to date in experimental techniques, point out to him the sig- nificant facts and data, and indicate the broad lines of theoretical inter- pretation. It was immediately apparent that the field of nuclear physics had grown so much and the various branches had become so specialized that no one person could hope to write a book like the famous treatises of Rutherford (which, however, because of the evolution mentioned above, had by 1930 already become Rutherford, Chadwick, and Ellis), Curie, and Kohlrausch. A cooperative effort like the Geiger-Scheel Handbuch der Physik seemed the only solution. Individual authors could undertake to prepare reasonably complete treatises on a restricted field in which they are quite authoritative. By keeping the discussions relatively short, it became possible for a group of authors to cooperate without cur- tailing their research activity. An incentive for several of the authors, indeed, was the desire to read the contributions of the others. This work is the outcome of that effort. We hope that it will be useful to the serious student and to the research worker in the field. Each part, with its bibliography, should be sufficient to inform the reader about the main results obtained in nuclear physics up to the end of 1952 and enable him to go directly to the original literature or to the several excellent collections of periodical reviews which are currently appear- ing (Annual Reviews of Nuclear Science, Reviews of Modern Physics, Ergebnisse der exakten Naturwissenschaften, Progress in Nuclear Physics, and others) for further details. E. Segre Berkeley, California June, 1958 CONTENTS Part VI. A Survey of Nuclear Reactions Philip Morrison 1 . The Conservation Laws 1 A. Application of Energy-Momentum Conservation. B. Conservation of Angular Momenta. 2. The Data of Nuclear Reactions 15 A. Yields and Cross Sections. B. The Measurement of Yields and Cross Sections. C. Types of Reactions: A Guide to the Literature. 3. The Nuclear Model 25 A. Qualitative Account of the Model. B. Calculation of Level Densities. 4. Nuclear Level Widths 45 A. Level Widths and Reaction Cross Sections: Statistical Relations. B. Calculation of Level Widths. 5. The Course of Nuclear Reactions 54 A. The Steps of the Reaction. B. The Contact Cross Section. C. The Disintegration of Compound States. 6. The Dispersion Theory: Resonance Reactions 64 A. The One-Body Model and Its Difficulties. B. The Dispersion Theory for an Isolated Resonance. C. The Generalized Theory of Dispersion: Many Levels and Many Decay Modes. D. Statistical Estimates. 7. Some Typical Nuclear Reactions 83 A. Resonance: The Region of Dispersion Theory. B. Reactions without Marked Resonance. 8. The Deuteron as a Projectile 110 A. The Oppenheimer-Phillips Reaction: Low-Energy Stripping. B. Strip- ping Reactions at Higher Energy. 9. Radiative Processes in Nuclear Reactions 114 A. The Multipole Classification. B. Calculation of Radiation Widths. C. Photo-Induced Reactions. 10. Nuclear Fission 123 A. The Energetics of Fission. B. The Products of Fission. C. Fission Cross Sections. 11. Nuclear Reactions at High Energy 141 A. The Nuclear Cascade. B. Correlations among Nucleons. C. The Op- tical Model for the Scattering of Nucleons. D. The Processes of Nuclear De-excitation at High Energy. E. Mesons: Virtual and Real. vii viii Contents Part VII. The Neutron Bernard T. Feld 1. Properties and Fundamental Interactions 209 A. Discovery. B. Properties. C. Fundamental Interactions. 2. Interaction with Nuclei 247 A. Introduction. B. General Considerations. C. Types of Neutron Re- actions. 3. Sources and Detectors: Neutron Spectroscopy 357 A. Introduction. B. Neutron Sources. C. Neutron Detectors. D. Slow- Neutron Spectroscopy. E. The Calibration of Neutron Sources. 4. The Interaction of Neutrons with Matter in Bulk 460 A. Introduction. B. Diffusion of Monoenergetic Neutrons. C. The Slowing-Down Process. D. The Nuclear Chain Reaction. 5. Coherent Scattering Phenomena with Slow Neutrons 508 A. Introduction. B. Neutron Diffraction and the Structure of Matter. C. Neutron Diffraction and the Determination of Nuclear Scattering Amplitudes. D. Magnetic Scattering and Neutron Polarization. Author Index 587 Subject Index 595 PART VI A Survey of Nuclear Reactions P. MORRISON Cornell University The study of nuclear reactions parallels the familiar study of chemical reactions. Most of the root ideas of chemistry are fundamental for the nuclear physicist. The equation of the reaction, the heat of reaction, the rate of reaction, the "balancing" of the equation — all these have their nuclear counterparts. The fact that the characteristic energy release is the millions of electron volts associated with the binding of nuclear particles, rather than the few volts which result from molecular binding, is the most striking inherent difference between the two fields. The nuclear physicist can detect the reaction not of moles or micromoles of his reactants, but of individual 'particles. This has meant that from the beginning the nuclear physicist has studied not equilibria and the laws of mass action, but the properties of single collisions. Just as the chemist has recently come to regard the individual collision as the center of attention for really fundamental understanding of reactions, so the physicist has now come to study large-scale reactions and even thermal equilibria in the nuclear domain in the light of his newer interest in astrophysical and chain reaction problems. Although the two points of view have tended to merge, it is our task to survey nuclear reactions primarily from the point of view of understanding individual collisions. Clearly this is the key to any genuine insight. SECTION 1. THE CONSERVATION LAWS Nuclear reactions, like chemical ones, can be more or less complex. The typical reaction — with which we shall be most concerned — involves a stationary or "target" nucleus bombarded by a relatively light incident nuclear projectile. The products of the reaction may be one, two, or even more nuclei. In the nuclear reactions of most interest before the development of the very high-energy accelerators in 1947, one to three products were by far most common. The target nucleus changes into a nuclear species differing by only a few units in A, the mass number or 1 2 A Survey of Nuclear Reactions [Pt. VI number of nucleons (protons and neutrons) in the nucleus, and in Z, the number of nuclear protons and, hence, the number of electrons in the neutral atom. Such reactions are designated either in the somewhat redundant notation taken from the chemists, an example of which might be uNa 23 + 1H 1 — * i2Mg 23 + on 1 [on 1 = neutron], or in an abbreviated scheme, due to Bothe. In the latter, one writes first the target nucleus, indicating its mass number and chemical character (often giving the nuclear charge Z as well), then in parentheses the symbols for the bom- barding and emitted particles in order, and finally the residual nucleus. The above reaction would be written Na 23 (p,n)Mg 23 . If there are more than two products of reaction, one might write, for example, C 12 (p,pn)C n for the reaction 6 C 12 + 1H 1 -» 6 C n + {H. 1 + on 1 . What physical properties of the system remain unchanged throughout the reaction? These conserved properties provide very valuable infor- mation for every reaction. We list conserved properties : 1. In no observed process does the total electric charge change. A proton can change into a neutron, but a positron or positive meson must appear, or an electron disappear, in the process. The familiar creation of electron-positron pairs is a dramatic confirmation of this principle. 2. To this date (1952) no process has been observed in which the total number of nucleons, i.e., the total value of A, is different on the two sides of the reaction. Presumably this is not a fundamental property of nucleons; the creation of positive and negative "proton" pairs is expected when sufficient energy is available. No such process has yet been observed. Nucleon number is conserved in all known reactions. 3. The constants of motion of classical mechanics are conserved, at least to an order of accuracy beyond experimental interest. These are total energy, momentum, and angular momentum. The energies of chemical binding, and the accompanying forces and torques, are negligi- ble compared to the energy transfers in typical reactions; so the colliding particles can be regarded as a mechanically isolated system. (For some special reactions involving very slow neutrons, the chemical binding and the thermal motion of the target nuclei must be considered. Even the whole of a macroscopic crystal may be taken as the target in some situa- tions. Compare Part VII, Section 5.) The conservation of total energy must of course include the energy equivalent of mass changes of the reactants, for this will generally be a large fraction of the available kinetic energy. The conservation of momentum guarantees that the incident and product nuclei move in a plane if only two product nuclei result. The total angular momentum includes of course the intrinsic or spin angular momentum of the reactants combined with the orbital Sec. 1A] The Conservation Laws 3 angular momentum of their relative motion. It is this total which must be conserved, though the breakdown of the total into intrinsic and orbital angular momenta may not be predicted in general. The experimental determination of the angular momentum of individual particles is nearly out of the question, but the statistical angular distribution of reaction products depends on the angular momentum relations. Measurements of such distributions are important aids to analysis of nuclear reactions. 4. The "constants of motion" peculiar to quantum mechanics are also conserved. These are two in number: the parity and the statistics (S3). 1 The parity refers of course to the behavior of the wave function of the system upon inversion of all coordinates in the origin, i.e., upon the changing of right- to left-handed axis systems. No physical property of an isolated system can depend upon the kind of axes used, but the wave function can change sign under such a transformation without affecting any expectation values. Whether the wave function changes sign or remains the same under such an axis inversion is indicated by the parity: even for wave functions remaining unchanged, odd for those changing sign. The parity of the initial wave function of the whole system must be conserved. The statistics of the system is related to these constants. The inter- change of all coordinates of identical particles in any system can lead only to change of sign or to an unchanged wave function. All particles must belong either to the class whose wave functions change sign when identical particles are interchanged (these are called Fermi-Dirac particles) or to the class whose wave functions must remain unchanged (these are called Bose-Einstein particles). From the relativistic theory of wave fields, it can be shown that particles with half-integer intrinsic angular momentum are Fermi particles; those with integer spin, Bose particles. In ordinary reactions, where the nucleons are simply rear- ranged, this conservation law leads to no new conclusions; but in proc- esses where new types of particles — mesons, neutrinos — are created or destroyed, this condition sharply limits the processes which can take place. A. Application of Energy-Momentum Conservation 1. Elastic Collisions. The simplest application of these principles is to the elastic collision of two free particles. The slowing down of neu- trons by elastic collision is the best-known example (see Part VII, Section 4), but, because the treatment is the basis for more complicated processes, we shall give the theory here, using the less familiar, but 1 References for the text of Part VI will be found on pages 189 to 192. A Survey of Nuclear Reactions [Pt. VI increasingly useful, relativistic form (B7). The non-relativistic formulas are of course contained in the low-energy limit. We begin with the target particle at rest in the laboratory coordinate system C. It is a simplification to make a transformation to a moving system C in which the laboratory and target particle are of course mov- ing, but the total momentum of the colliding particles is zero. Lorentz transformation from this system to C will give us the results we want; we know that the energy-momentum conservation is automatically satisfied in any Lorentz frame if it holds in C". Let the target particle of rest mass m 2 be stationary in C, and the other particle of mass mi be incident with velocity v along the x axis. In the system C" the particles approach each other with velocities «/ and u 2 respectively. We require the particles to have the equal and oppositely directed momenta p' before collision. Then V = miUi m 2 u 2 (1-fr' 2 )^ (1-/J 2 ' 2 ) M where j3' = u'/c. Using the notation 7' == 1/(1 — /3' 2 ) M , we have (la) Squaring, c «i 8 W 2 - 1) = m 2 2 (y 2 ' 2 - 1) (lb) (2) Since the total momentum remains zero after the collision, the final momenta are easily seen to be equal and opposite, but the particles now are receding instead of approaching. Compare Fig. 1. If particle 1 Fig. 1. Momentum relations for collision in center-of-mass system. is scattered through the angle 6' in C", particle 2 will come off at an angle w — 6' with respect to the initial direction of particle 1. The velocities are not changed because the kinetic energies are conserved in an elastic collision. Sec. 1A] The Conservation Laws 5 We now make a simple Lorentz transformation to the laboratory rest system C. If particle 2 is initially at rest in C, C must be moving relative to C with velocity u 2 in the direction of the incident motion. Now we know that a four-vector is formed by the momentum components and the total energy E/c 2 , transforming under Lorentz transformation like the coordinates x, y, z, t. Therefore the Lorentz transformation gives for the initial momentum of the first particle in the system C" (omitting a factor of m\C throughout) : (7/ 2 - 1) K = 7 2 '[(7 2 - 1) K ~ /Vy] = 7 2 '(7 2 - 1)* - (7 2 ' 2 - 1) M 7 (3) where m,ic(y 2 — 1)^ is the momentum, and miC 2 y the total energy of particle 1 in the laboratory system C. The kinetic energy is m\C 2 (y — 1). Using (2) and (3), we find ■ -2/ Ti = 7 : , .^u (4a) 72 = 1 + (4b) To obtain relations between the scattering angles, we again use the fact that the momentum and energy E, divided by c 2 , form a four-vector. The total energy E 2 of particle 2 after the collision and its momentum components, in the laboratory system, are given by E 2 = 72 W - c/SaV cos 6') (5a) Px 2 = 7 2 y (-p' cos 0' + - /9 2 'tf 2 'j (5b) P»2 = -P'sinfl' (5c) where the incident direction is the x axis, and all the momenta lie in a plane. The energy of particle 2 in C is E 2 ' = m 2 c 2 y 2 ', and its momen- tum p' = m 2 y 2 'c(3 2 '- Dividing (5b) by (5c), we find the angle of scatter- ing $2 of particle 2 in the laboratory system C: 1 1 tan 9 2 = -cot - 6' (6) 72' 2 6 ' A Survey of Nuclear Reactions [Pt. VI The minus sign means that, if particle 1 is scattered above the x axis into the region of positive y, particle 2 recoils below the axis. If 0' is 180° in the center-of-mass system C, 6 2 = and particle 2 recoils for- ward with maximum energy. As 0' decreases toward zero, particle 2 recoils with decreasing kinetic energy at angles approaching 90°. The case 0' = corresponds to grazing collision, in which particle 1 loses no energy and particle 2 recoils with zero velocity at 90°. The recoil kinetic energy of particle 2 in the laboratory system is 7' 2 = E 2 — m 2 c 2 . From (5a) T 2 = m 2 c 2 (y 2 ' 2 - 1)(1 - cos d') (7a) Using (6) and (4b), we can write this in terms of the angle d 2 . Finally we may introduce the total energy E = m,\<?y and momentum p = mic(y 2 — 1)^ of the incident particle, getting p 2 COS 2 2 T 2 = 2m 2 c 2 (7b) (E/c + m 2 c) J — p 2 cos 2 2 Evidently 2' 2 (max) = 7' 2 (0 2 = 0). It is of interest to compute the maximum fraction of the kinetic energy of the incident particle which can be transferred. T = E — m\C 2 = mic 2 (T — 1), and 7V max) _ 2( 7 + \)m x m 2 _ 2( 7 + \)m x m 2 T m 2 + 2ymxm 2 + m 2 2 2(7 + l)mim 2 + {mi — m 2 ) 2 This fraction increases monotonically with incident energy from the value familiar in the non-relativistic limit, 4m 1 m 2 /(mi + m 2 ) 2 , to unity for very high energies, with 7 » 1. If the particles have equal mass, the maximum recoil energy is of course always equal to the incident energy. If the particle incident is light (e.g., electron or photon), particle 2 can acquire relativistic energies only if p ■= m 2 c 2 or 7 = m 2 /m\. This is also the condition for transfer of a large fraction of energy from par- ticle 1 to particle 2, if a heavy particle is incident on a light target particle. We now discuss the relation between the scattering angle 0i of particle 1 and its energy loss. The total energy of particle 1 in the laboratory system C corresponding to a scattering through 0' in C is E x = m lC 2 [7i'72' + (7i' a - 1) } ^(72' 2 - 1) H cos 0'] (9) where the 7"s are as given by (4a) and (4b). Now we need cos 0' in terms of the angle measured in the laboratory system. For this we Sec. 1A] The Conservation Laws 7 apply the Lorentz transformation to the energy and momentum of particle 1 in the manner of (3). This shows 1 sin 0' tan 0! = (10) y 2 (cos 0' + TO 1 Y 1 '/W12Y2') Solving for cos 0' in terms of tan 0\, we get a quadratic equation with two roots, indicated by the ± sign: cos 0' = mi , 7l ' T2 ' tan' 5 0x (1 + T2' 2 tan 2 X ) L m 2 2 ™ 2 1 5 tan 2 ^) (1-1) Since the energy for a definite scattering angle in the laboratory system is given by (9) in terms of cos 0', it appears that there are two possible values for the energy of a particle elastically scattered through a definite angle. It is instructive to examine the situation graphically. For non-relativistic velocities (7 = 1), Eq. (11) gives the tangent of the scattering angle in C in terms of functions of the angle 0' in C". For higher velocities, we can define an auxiliary angle ^ by the relation tan \p = y 2 tan0j. This angle \f/ can be obtained geometrically by adding a vector of length m^ix /m 2 y 2 along the x axis to a unit vector whose components are cos 0' and sin 0'. The center of a unit circle represents C", and the scattered particle may go off in any direction, along a unit radius vector. The auxiliary angle ^ is then just the angle between the x axis and the resultant of the radius vector in C added to the vector miyi'/m 2 y 2 '. The construction for the three cases we must consider is shown in Fig. 2. In case (a), with mi > m 2 , there are two possible directions of scatter- ing in C" (i.e., two distinct orientations of the unit vector) for one scatter- ing angle 0i in C. There are thus two values for the energy of particle 1 , in agreement with (11). For the case mi < m 2 , Fig. 2b is appropriate. Here the a>axis vector is less than unity, and the construction gives only a single angle 0' to be associated with a definite i/'. The analogue to the second orientation of the unit vector in case (a) corresponds to the angle (x — \f/) and gives no additional solution. One of the roots of (11) is to be rejected if mi < m 2 . From the diagram we can see that, if 0i is less than 90°, we should take the more positive value of cos 0' in (11); for X greater than 90°, the more negative one. 8 A Survey of Nuclear Reactions [Pt. VI We shall treat case (c), mi = m 2 , as a limit of either (a) or (b). From the figure we can see that there is an upper limit to ^ and thus to the angle 1; corresponding to the case where the line defining \p is tangent (a) m 1 >m z Unit circle (b) m 1 <m 2 (c) m^Mj Fig. 2. Scattering angle relations for elastic collisions. Incident particle is mi; target particle, m^. The angle 41 determines the laboratory scattering angle. See text for construction. to the unit circle, and angles 0'(u, 0\ 2 ) have coalesced to a single value. The radical in (11) must vanish at this maximum value: tan^""" = m 2 m\ — m 2 (12) 9i (max) = arc s i n m 2 nil In case (b), with m\ < m 2 , scattering can clearly occur for all angles between and t. In the limiting case (c), mi = m 2 , the condition of tangency is reached for ir/2. In the special case (c), all the formulas above are much simplified. From (4a) and (4b) we get Ti' = 72' = {—£-) ( 13 ) Sec. 1A] The Conservation Laws 9 If T* is the kinetic energy of the particle scattered through angle 6 in C, either (5a) or (7b) gives 2(7 - 1) cos 2 (7 + 1) ~ (7 - 1) cos 2 and, with T = mc 2 (y — 1), T cos 2 e T* = mc 2 — (13a) 1 + f (7Vmc 2 ) sin 2 mi — m 2 = m (13b) Although there are still two roots of (11), one of them always corresponds to 0' = 180°. This means of course that one of the colliding particles is projected in the forward direction with the full incident energy, while the other remains at rest. The second root of (11) gives (13a). The familiar non-relativistic form of (11) is easy to obtain. If »ti = m 2 , the scattered particles always come off at right angles to each other, with energies proportional to the squares of the cosines of the angles of deflection. For higher velocities, however, the angle included between the directions of the outgoing particles is always less than a right angle. Changing the sign of 2 in (6) and using (10) and (13a) and (13b), we find 2 tan di tan 2 = (14) 1 + 7 Since y is always greater than unity, 0i + 2 is less than 90° except for the grazing collisions in which particle 2 may be thought of as scattered at 90° with zero velocity. A sufficiently accurate measurement of angles 0i and 2 would serve to determine the incident energy. This method is more sensitive for large values of y. Finally we set out the familiar relations for the non-relativistic case. If T 2 is the kinetic energy of recoil of particle 2 and T is the incident kinetic energy of particle 1, T 2 =- -^cos 2 d 2 T (15) (mi + m 2 ) i For the final kinetic energy of particle 1, from the proper limit in (9), we obtain \mi + m 2 / L \m\/ /m 2 2 . y ± 2 cos 0i I — r — sin 2 0! I The treatment of the two roots is the same as in Fig. 2. (16) 10 A Survey of Nuclear Reactions [Pt. VI 2. Collisions with Creation of New Particles. For sufficiently high- energy collisions of nuclear particles, not only are rearrangements of nucleons possible, in which kinetic energy is transformed into binding energy (or the reverse), but also new particles may be produced whose rest energy comes from the kinetic energy of collision. In a collision in which additional particles are produced whose rest energy totals Mc 2 , it is clear that energy conservation requires that at least the incident kinetic energy exceed Mc 2 . In addition, because the conservation of momentum requires that the center of mass of the whole system move with un- changed momentum, we must usually provide additional energy to fulfill this condition. Let us consider the threshold value of the kinetic energy of the incident particle, ra x , on a stationary target particle m 2 (F8). The threshold is the minimum energy below which the production cannot take place. How probable the production becomes just above threshold is of course impossible to tell in general; it will depend upon the particular inter- actions being considered. It is easy to see that at the energy of threshold the total energy of all particles is a minimum after collision. In the frame C" we know that the final momentum must be zero. These two conditions can be satisfied if all particles after collision are individually at rest in the center-of-mass system C". Then the initial kinetic energy must have been just equal to the increment in rest mass energy, Mc 2 . The total energy in the C" system before collision is E t ' + E 2 ' = mi c 2 yi ' + m 2 c 2 y 2 ' and from (4) we can write this Ex' + E 2 ' = c 2 (w! 2 + 2ym } m 2 + m. 2 2 )' A (17) Equating it to the energy after collision, Ei' + E 2 ' = c 2 ( mi + m 2 + M) Solving (17) for y, we obtain the threshold kinetic energy T of mi in the laboratory system from the relation T = m A c 2 (y — 1). This gives Mc 2 {m l + m 2 + IM) T = LJ *—L (18) m 2 To create a nucleon pair from a proton-proton collision, for example, would require T at least 2 ■ m p c 2 (3m p /m p ) = 6m p c 2 . If the pair were made by a gamma-ray or electron incident {m x ^ 0), the threshold would be T = 4m p c 2 . In such collisions the target particle need not be at rest. Here we may think of a target nucleon as one of the nucleons bound in a nucleus, Sec. 1A] The Conservation Laws 11 and thus moving with the velocity characteristic of its zero point vibra- tions. If the collision occurs when the target nucleon is moving with a velocity component directed toward the incident particle, less incident beam energy will be needed. If the target particle m 2 has the initial velocity v 2 in the x direction, Lorentz transformation shows that the threshold energy of (18) becomes reduced to the value T min = y*(T - v 2 p) + m lC 2 ( T2 - 1) (19a) where p is the initial momentum of particle mi, and 72 = (1 — v 2 2 / c 2 )~ V2 . A further reduction comes about if the collision occurs with capture of the incident particle so that binding energy can be released. It is of some interest to discuss the possible angular and energy dis- tribution of the produced particles. If we restrict ourselves to the case in which the target particle is at rest, and in which only one new particle of mass M is made, we can draw some simple conclusions. It is clear that at the threshold energy there is only one final condition: all the particles move forward with the same velocity, since they are all at rest in the center-of-mass system. As the energy exceeds the threshold, the created particle will in general be able to travel in a distribution of angles around the forward direction. The details will of course depend on the nature of the interaction. But some limits can be given in general. It is clear that the maximum momentum of the new particle will corre- spond to a case in which M is traveling forward and the two original particles move opposite to M in the center-of-mass system, with the same speed for each. This leads to a value for the maximum kinetic energy of M in the laboratory system : Mc 2 + r m « = \c 2 [{ynii + m 2 )A + mi(y 2 - \) }i B]/{mi 2 + 2ym 1 m 2 + m 2 2 ) (19b) with A = M 2 + 2(t - \)m x m 2 and B = {[2(7 - l)m 1 m 2 - M 2 ] 2 - AM 2 (nix + %) 2 ! ^. This has a very simple form at the threshold: Mc 2 Mini! + M/2) T max = — (20) m 2 mi + m 2 + M The angular distributions will be controlled by the relation between the maximum velocity with which M can move off in the center-of-mass system and the velocity of C with respect to C". Arguments similar to those accompanying Fig. 2 show that here too there are two possibilities : the new particle will be emitted only in part of the forward hemisphere if its maximum center-of-mass velocity vm = c 2 pm'/Em' is less than the velocity of the center-of-mass system, c/3 2 '. Then the situation is 12 A Survey of Nuclear Reactions [Pt. VI like Fig. 2a, and a minimum energy for M in the laboratory exists; it is given by (19) with a minus sign instead of a plus sign before the radical. If v M ' > c/V, however, the case is like that, of Fig. 2b, and all angles of emission are allowed in the laboratory frame. The minimum energy for M is then zero in the laboratory system. 3. Reaction Energies. The most studied nuclear reactions up to the present time involve neither elastic collisions nor the production of new particles. They consist of the rearrangement of nucleons, with the ejection of different nuclei having correspondingly changed binding energies and rest masses. Just as in chemistry, the heat of reaction Q is a significant quantity. The energy Q is the energy released as kinetic energy (or the energy of photons, etc., set free) at the expense of the internal energy of the colliding systems. We may define Q by either of the relations: Q = c 2 ( £ tru -EmA = -I Ti + £ T f (21) V initial final / initial final where the notation is that of Section 1A2, and the equivalence of the definitions follows from the conservation of total energy; T { is defined as kinetic energy. We consider first the case of a typical reaction of the two-product type: T(i,p)R. It is very easy to treat this case with the target nucleus at rest. Then conservation of momentum in the laboratory system yields the vector relation P; = p p + Pr. We have only to square this, employ the relativistic connection between momentum and kinetic energy T, p 2 = T 2 /c 2 + 2mT, and substitute in the definition for Q, eliminating, say, T R . We have \ 2 T Q = T P -Ti + m R c 2 + m R / m, X T T IrripC ■ m-iC /m p V T p / [ T p \ \tor/ m p c 2 \ m p c 2 / -(— +2)- { C 2 \TOjC 2 / m p &/ \ rriicr/ m p c 2m p nii ra R 2 - 1 (22a) This may be written in the low-energy limit by expanding the radical, neglecting T/mc 2 compared to unity for all particles. The familiar result follows: m / m v\ ( w,- \ (m v T v mi'. Q = T p [l+—)-Ti[l -)-2 P P \ TOr/ \ Tor/ Tor miTi)> Vz cos 6 P (22b) Sec. IB] The Conservation Laws 13 In reactions of this type, with only two product particles, there is evi- dently a unique value of T p for a given recoil direction and a given inci- dent energy. When the reaction produces three or more particles, no such simple connection exists. The case is then parallel to the discussion in Section 1A2. A distribution of energies results. If the reaction takes • place in two independent stages, first the emission of two particles, then the break-up of one of the pair, somewhat simpler relations can be obtained. Thus, study of the energy distribution can give some insight into the mechanism of the reaction. The rigid correlation between energy of outgoing particle and angle of emission has been much exploited as a laboratory means for obtaining particles of a well-defined energy. For neutrons especially such mono- energetic beams can hardly be obtained in any other way. Simply as an example, it is interesting to note that the very exothermic reaction T 3 (d,w)He 4 used with a well-controlled 1-Mev beam of deuterons pro- duces neutrons ranging from almost 16 Mev to under 13 Mev in the backwards direction. A study of a set of such reactions has been made which makes possible the production of monoenergetic neutron beams of energies from a few kilovolts up to about 20 Mev; a source of charged particles of well-defined but rather modest energies of a couple of Mev was used. A valuable review by Hanson et al. (H2) fully discusses the several reactions used most frequently and gives a graphical treatment of the fundamental relations. Graphical treatments of the relativistic case, very useful when it is desired to reduce experimental data, have been given in (B16) and especially in (Ml). It is worth while to make the remark in closing that, in all the relati- vistic formulas above, the case in which some particle (i) is a photon can be obtained by replacing the total energy Ei = mic 2 y by the quantum energy hv, and neglecting m,- wherever it occurs without a factor 7. B. Conservation of Angular Momenta The complementary relation in quantum mechanics between energy momentum and space-time leads to straightforward experimental use of the ideas of energy and momentum, as we have seen. The important canonical variable, angular momentum, is conjugate to angle. We measure the angular momenta of quantum systems mainly by obtaining statistical distributions in angle. The typical nuclear reaction is carried out with a beam of particles ; the very term beam implies a more or less sharp definition of the direction of motion, and hence a necessarily rough limitation on angular momentum. An infinite plane wave, indeed, contains all angular momenta of orbital motion, with definite ampli- tudes, and in addition requires supplementing by the wave function 14 A Survey of Nuclear Reactions [Pt. VI factor representing the intrinsic angular momentum of the particles in the beam. Only when there is some reason to limit the states of angular momentum which are of interest — either simply formally in some systematic counting of all angular momentum states (as in the method of partial waves) or physically because for some reason only a limited number of angular momenta contribute to a given reaction — can the angular momentum conservation law be of much value. For example, in the familiar case of thermal neutron reactions, where the incident wavelength is large compared to the region of possible interaction, only the spherically symmetric component — angular momentum equal to zero — of the incident wave can possibly contribute. Other parts of the wave are of vanishingly small amplitude; the "centrifugal barrier" keeps them from the region of interaction. If any specification can be made of the angular momentum (hence of the variation with direction) of the incoming wave, general rules may often become helpful. The total angular momentum is certain to be conserved; the forces between nucleons are, however, non-central, so that there is a tendency to interconvert intrinsic or spin angular momentum with that associated with orbital motion. Only in special cases can the relative apportioning and hence the complete angular distribution be obtained (Y2). The spin angular momentum affects, of course, not the direction of motion in space but the orientation of the spin axis of the particle concerned. If an unknown spin change can occur, this will clearly affect predictions of angular distribution in a way calculable only under further specifica- tion of the interactions involved. There are three general results, inter- esting to present, which apply quite generally to all nuclear reactions between unpolarized target and unpolarized incident particles: 1. As is clear on physical grounds, there will be axial symmetry about the direction of the incident beam. 2. If, among the incoming partial waves, only those of angular mo- mentum L or below contribute appreciably to a reaction, the angular distribution of any single product particle cannot be more complicated than that of the incoming contributing partial wave. Indeed, the angu- lar distribution of the outgoing particle will be a polynomial in cos 6, where 6 is the polar angle of emission relative to the beam direction, of degree no higher than 2L. This holds independently of the spin of any particles or of the number which take part. It is restricted to non- relativistic velocities for the incoming particle. In relativistic cases, one higher power of L may occur, but in general it will be reduced by a factor of the order of (v/c). If several partial waves of different L con- tribute, the polynomial will generally contain all powers of cos 6 from Sec. 2A] The Data of Nuclear Reactions 15 the maximum down; if only one wave contributes, only even powers of cos 6 can occur. These results refer, of course, to the center-of-mass frame of reference. 3. If the distribution of product particles contains any odd powers in cos 8, i.e., if it is not symmetrical with respect to the plane normal to the beam, two (or more) intermediate states are involved with opposite parities. Thus an even polynomial in cos 9 is a typical consequence of reactions involving a marked resonance; this will be discussed more fully later. Most frequently such considerations yield useful information only when there is additional information, for example data permitting the original specification of what partial waves can contribute. This implies some statements about the mechanism of the reaction, some definite nuclear model, and some assumption about the character of nuclear states and of interactions. Then the selection rules which result are often powerful discriminants between possible alternatives. Such cases occur most frequently in nuclei lighter than neon, in which region some progress has been made in constructing detailed models for nuclear systems. Examples will be given in the proper places later. SECTION 2. THE DATA OF NUCLEAR REACTIONS The conservation laws are satisfied in every single nuclear collision. But, besides data on the constants of motion, nuclear physics also is concerned with knowing how many nuclear processes of a definite kind take place under given conditions. It is evidently beyond us to say whether or not the next proton, say, will initiate a transmutation in a given fluorine nucleus, but we can obtain from both experiment and theory the probability of the process. We can expect to predict with calculable accuracy the fraction of the beam of incident protons which enters the target nuclei. A. Yields and Cross Sections The simplest expression for the probability of a nuclear reaction is the yield. This is frequently but a semi-empirical expression, stipulating, for a particular experimental arrangement, how many processes occur per incident particle. The statement that a yield of 1/2000 was obtained for the reaction Li 7 (p,a)a with a thick target at 3 Mev means only that one lithium nucleus divides for every 2000 protons stopped in the target material. The specification is obviously incomplete. For data about the probability of the reaction itself one would need to know how many 16 A Survey of Nuclear Reactions [Pt. VI encounters occurred and at what energies, whereas here the protons lose energy as they penetrate the target, which is often not even a single nuclear species but a mixture of isotopes or even a compound. Improve- ment could be obtained by using a target so thin that the energy loss of the beam protons in traversing it was negligible ; here at least the energy would be sharply defined. Thin-target or thick-target yield data are common in the literature; each requires special interpretation before any absolute number can be obtained (CO, Rl, R2). Such a form of presenting data implies that only some relative feature of the reaction is under study. The familiar cross section is a complete specification of the probability of nuclear reaction. A cross section oab for a reaction in which the par- ticle A is incident with specified properties and the specified set of particles B emerges can be defined by the expression n# = Ia^abN. Here N is the number of target particles presented to the beam (for example, for thin targets the product of beam area, target thickness, and number of target atoms per cubic centimeter gives JV), n# is the number of the specified particles B emitted per second, and I a is the incident flux of particles A, the number per unit area and time. 1 A cross section may be specified for any process or partial process: for example, we can assign a cross section for the emission of gamma-rays of energy E in the direction specified by the polar angles (8, <p) when protons of energy W are incident on Li 7 . Then the number of such gamma-rays emitted in that direction per unit solid angle and per unit energy for unit proton flux is just <r py (E, 6, <p) for each lithium nucleus in the beam. The quantity a P1 is called a differential cross section (per unit energy and unit solid angle), since it is defined for an infinitesimal range of the continuous variables E, <p, and 8. The total number of gamma-rays emitted in all directions is of course obtained by summing over all energies and directions. Then the number of gamma-quanta for unit flux and a single target atom is S P7 = / dE I dip I d(cos 8)<r py (E, <p, 8) Jo Jo J +1 Here 2 PT would be called the total cross section for proton capture by lithium for protons of energy W. In this way the frequency of any process or partial process can be described fully by giving the appropriate cross section, which may depend on any set of discrete and continuous 1 Note that for the collisions of identical particles, as in the scattering of protons on protons, incident and recoil particles are not physically distinguishable, and the number of observed events may then be a sum of both. Sec. 2B] The Data of Nuclear Reactions 17 variables. The cross section for any process is just the area of the inci- dent beam from which particles are removed by the given process, when the beam is thought of as directed at a single target nucleus. Of course, the cross section may be denned not only per nucleus, as we have here done, but also per unit volume, per gram, or in terms of any convenient measure of target atoms. If there were no non-classical wave effects, the cross section for a definite target, summed over all the possible processes which could occur, would be just the geometrical area of the target. We shall see almost this result for the removal of fast neutrons from a beam (B14, L2, M3, W5). The probability of a nuclear reaction, expressed either as a cross section or only roughly as a relative yield, is most frequently measured for a specific nuclear reaction as a function of the energy of the incident beam. Such a number is called the excitation function of the reaction concerned. (If a change in chemical identity is involved, it is sometimes called a transmutation function.) The chief data of nuclear experiments besides the energy relations and the nature of the reaction are the angular distributions and the excitation functions, which can be reduced to a knowledge of the cross section for the specified reaction as a function of energy of incidence and direction of emission. B. The Measurement of Yields and Cross Sections The experimental procedures for the measurement of cross section amount to a good fraction of the subject matter of experimental nuclear physics. It would be presumptuous to try to discuss them in a small space. It seems worth while, however, to give a kind of summarized functional enumeration, not so much of the experimental methods, but of the general procedures and precautions which must be included in any program for the measurement of cross sections. Evidently the type of measurements made will depend on the detail and the accuracy of cross- section knowledge wanted; it is obviously more exacting in general to seek a knowledge of the differential cross section than of the total cross section, and of an accurate absolute value than of a rough relative one. 1. The Beam. If the cross section a-,-, t(E, 6,<p) is sought as afunction of energy, of angle, of type of projectile, it is obvious that the beam must be well defined with respect to these quantities. Usually this definition is imposed on the entire incoming beam of projectiles; some- times it is possible to study the reaction products and by some condition of the reaction eliminate all events which did not begin with the correct value of one of the variables. These methods frequently involve time- coincidence counting. Generally the entire beam accepted into the re- gion of reaction is known to have the wanted values of its parameters. 18 A Survey of Nuclear Reactions [Pt. VI The type of projectile is the most general variable which needs control. The emerging beam from a cyclotron is likely to be quite free of all types of ions other than the resonant species, but an electrostatic generator, for example, will equally well accelerate the deuterons and the protons which may leave its hydrogen-ion source. Molecular ions contribute to the current like atomic ions, but not to the yield, and must often be specifically excluded. For such purposes magnetic resolu- tion of the raw particle beam is often used. When unstable particles form the beam, as in the case of pi-mesons, any stretch of beam travel will allow decay particles to enter the beam. Such beams cannot be wholly pure, and corrections for the mixture of particles are needed. Usually charged particle beams are rather easily controlled for beam purity because of the power of ion-optical analysis. Gamma-ray beams, or beams of charged particles energetic enough to yield neutron-emitting reactions in any intervening windows, air, etc., in the path, may require special care. Crude magnetic analysis will generally remove electrons; neutrons are much harder to eliminate and may cause trouble. Deflec- tion of the wanted charged beam will in principle solve the neutron problem, if scattering from the magnets, etc., is not too severe. The problem is one of increasing importance as beam energy, and with it the number of secondary reactions, increases. The energy is often very well defined by the source used. Control of d-c ripple in electrostatic generators has reached precisions of a few parts in 10 s . Magnetic analysis can be pushed nearly as far even for roughly defined beams like the cyclotron output. Synchrotrons give excellent control over beam energy as well. Any matter which the beam traverses in its path to the target produces energy losses by ionization and excitation of atoms, and also by nuclear events, which smear out the beam energy. This is in principle avoidable by keeping the beam in vacuum, but eventually the beam must strike some target material. The same processes, of course, occur in the target, which must be kept thin (measured in atoms per square centimeter) if the cross section is wanted at a well-defined energy. Thick targets can yield only cross sections integrated over energy, and may cause straggling in direction and even particle type as well. Yield as a Junction of energy sometimes can roughly replace beam homogeneity by allowing a differentiation of the integral yield. Homogeneity in energy usually is easier to secure than absolute knowledge of the energy of the beam employed. The energy scale in the region of Mev depends on comparison with a few nuclear resonance energies which have been calibrated absolutely by measuring the trajec- Sec. 2B] The Data of Nuclear Reactions 19 tory of a proton beam in a known electric field. 1 This gives an accuracy of about one part in a thousand. Magnetic trajectories using the proton magnetic moment resonance frequency for field measurements work about as well, and extend into the 100-Mev region, especially for elec- trons from gamma-ray beams. Range measurements using the theory for energy loss by ionization are very valuable secondary standards over all energies above, say, a few kev. They have been computed and calibrated over this whole range for nearly all the usual projectiles. The angle of emission of the Cerenkov electromagnetic shock wave in a material of known index of refraction is an elegant method for particles with a velocity such that j3n > 1, where n is the refractive index of the material; velocity measurements by direct time-of -flight determination have also been made. The kinematics of elastic collisions is also capable of giving beam energies expressed as a ratio to the rest energy of the particle, in the relativistic region. Angle definition for the beam, often called collimation, is of impor- tance for any measurements of dcr/dQ. Here defining slits of some sort are the usual key parts of the apparatus. Sometimes the direction of the track of the product particles themselves, made visible by cloud chamber or nuclear emulsion techniques, replaces slits followed by direction-insensitive counters. The effects of Coulomb scattering from nuclei and electrons of the target itself or any material in the beam path are the main enemy of good collimation. Divergence of the original beam forms in itself no source of error which collimating slits or simple distance of beam travel and choice of detector acceptance angle cannot remove. But good collimation implies loss of total beam intensity, and practical limits of time and background here often are decisive. Perhaps the most important of all parameters of the beam is the number of particles striking the target in a specified experiment. Often the current is measured and then time-integrated. Monitoring, or con- trol of relative intensities, is frequently used for a series of experiments in which absolute values are either ignored (e.g., all cross sections may be measured relative to a given known cross section) or calibrated against one single measurement. Integrated charge may be measured 1 Typical reference energies in absolute volts are F 19 (p,a T )0 16 0.3404 Mev Resonance Li 7 ( P)T )Be 8 0.442 Resonance F 19 (p,q:t)0 16 0.8735 Resonance AF(p, T )Si 28 0.9933 Resonance Li 7 (p,n)Be 7 1.882 Threshold See, for example, W. Hornyak, T. Lauritsen, P. Morrison, and W. A. Fowler, Revs. Modern Phys., 22, 291 (1950). 20 A Survey of Nuclear Reactions [Pt. VI by measuring the potential change or integrated current flow to a Fara- day cup, an insulated conductor so arranged that no secondary charges can leave it and all beam charge is stopped within it. In one high-energy proton experiment the Faraday cup was a forty-pound chunk of brass under magnetic field in a vacuum. More usually it is a small cylinder with a beam window in one end, Ionization is often used to measure intensity by taking advantage of the known space rate of energy loss and the mean energy used per ion pair. Direct counting of ionizing tracks in photoplates can be used. Relative values can be obtained by actually measuring the beam-induced yield of some known reaction, using radioactivity, particle emission, or even collecting the neutral atoms of a stable reaction product, as has been done for such reactions as IA 6 (n,H s )a. Even calorimetry of a target which stops a totally absorbed beam has been used for beam intensity. With particles of non-zero spin (protons or even gammas) there is another possible parameter to control: the; spin component. This cor- responds to use of a polarized light beam in optical experiments. Apart from magnetic polarization of neutrons, only the slight polarization effects resulting from double scattering have been examined for charged beams. This may become a more important field in the future. 2. The Target. To specify a cross section precisely, the nuclear composition of the target must be well known. This implies pure and homogeneous targets. The isotopic concentration of the target must, of course, be known ; this is usually easy if natural targets are used, but special analysis is required with targets prepared from isotope-separated material. The target nucleus may undergo the looked-for reaction, but in gen- eral this is improbable. Most of the target nuclei are not changed by the beam. Their presence is needed because the probability of reaction is proportional to the number of nuclei independently exposed. But their very presence modifies the beam by producing straggling in energy and direction by Coulomb scattering from electron and nucleus, or by occasional nuclear collisions of kinds other than that under study. They may produce a "straggle" in beam particle type by yielding secondaries which go on in the target to induce unwanted reactions of nuclei later in the beam path. They clearly modify the beam intensity. Obviously the use of targets sufficiently thin that the changes in any of these properties is small compared to the resolution desired in each parameter is always a solution. But it may be impossible to employ such a target because the total number of reactions then becomes too small, or because the physical form of the target elements prevents preparation of a thin sample. Then a careful study of the beam be- Sec. 2B] The Data of Nuclear Reactions 21 havior in the target material must be made, and corrections made for all the straggling effects. A frequently employed scheme of finding cross-section variation with beam energy is to use a target made of a stack of thin foils, the yield in each of which is measured. The energy variation of the beam with target thickness tells the mean energy in each foil, and thus the excitation function is measured. The scheme demands that the range of the product particles in the target material be less than the range intervals used in choosing the foil thickness; otherwise the yield function is partly displaced in a distance and energy scale, and detail is lost. Experimental problems of great importance sometimes arise from the heat dissipation and other physical effects of strong ionizing beams. Constancy of the target mass and composition is assumed in nearly all measurements; evaporation, diffusion, coatings from soot of decom- posed pump oils, mechanical displacements of target material by melt- ing or otherwise, and radioactive recoil must all be avoided or corrected for. 3. The Measurement of Cross Sections. The most direct method of measurement is an initial absolute measurement of some differential cross section which defines experimentally the angular acceptance, particle type, energy, etc., of the reaction product. Then a count of the number of such events from a fixed target thickness (with correct- tions pointed out above) and with known beam population yields the cross section directly. The means employed are diverse. (a) Measuring the Residual Nucleus by Detecting Radioactivity. This may be connected with a radiochemical separation of that nucleus from the target matrix. Such a scheme does not define the energy, angle, spin, or any other property of the product nucleus except its chemical, or indirectly its nuclear, identity. It may be combined with the thin- layer target scheme to get excitation functions for a specified reaction. (b) Collecting the Product Particle. If this is done, as usual, while the product particle is in flight, the angle, energy, and type of product may all be defined. Very often ion-optical means are used. Such schemes separate product particles by their differing trajectories in a vacuum chamber with a controlled electromagnetic field and make possible detection of individual levels of the residual nucleus by careful meas- urement of the energy released. The charge-mass ratios usually fix the product uniquely. Detectors are used to count the product particles, which may be linearly responding ionization chambers or solids giving scintillation response, gas-amplifying counters like the proportional and Geiger counters, photoemulsions, cloud chambers, simple charge collec- tion by Faraday cup, or specific induced nuclear reactions themselves 22 A Survey of Nuclear Reactions [Pt. VI (W10). The angular resolution is evidently mainly a problem of the disposition of the apparatus. Energy resolution may be secured by such ion-optical means as we have mentioned, or by using absorbers to fix the total energy needed before particle counting. Counter telescopes which require energy loss rates to be within a fixed range, determined by signal levels and the coincident or anti-coincident response of a fixed set of counters and absorbers, are very much used at the high-energy end, where ion optics is more difficult. Cerenkov counters and time-of- flight measurements are among the newest techniques coming into use in this field. It is possible to measure total cross section, a = I dil d<r/d&, more easily. Here what counts is removal from the beam. All that is needed then is a comparison of detector response with and without the inter- vening target. For thin targets having N atoms/cm 2 and "good geom- etry," then, I/I = e~ N ". "Good geometry" means such collimation that all particles scattered through any small angle miss the detectors (B14). Since the detectors have finite acceptance in every case, correc- tion is needed for scattering-in (P3). When this correction is sizable, the case is said to be one of bad geometry. Thick targets, where multi- ple collisions are not negligible, require additional care. The inescapable relation in Eq. (45) between an absorption cross section and elastic scattering means that special care must be taken to correct for the very low-angle forward coherent scattering if the geometry is very good, since scattering-in cannot be estimated from the angular distribution of the main absorptive process. Apart from such questions, it is in general possible to make the only absolute cross-section measurement one of total cross section, done by beam attenuation, and then only relative measurements for all the differential cross sections are needed. 4. Special Problems with Gamma-Rays. The use of gamma-rays as beam projectile, or their measurement as product, has special problems arising from the small interaction with matter. Gamma-rays can be detected only after they have given their energy at least in part to mat- ter, especially to electrons. For this purpose most gamma-ray-detecting schemes make use of a layer of matter of appropriate thickness placed ahead of the detector proper. In this converter layer the gamma-rays produce electrons (or positrons) whose ionization, range, energy, and momentum may be measured by the means described above. At ener- gies in the range of a few Mev it is possible to get absolute measure- ments of gamma-ray flux by using the known ionization of the electrons secondary to gammas, which of course have a spread of energies as long as Compton scattering is important. At energies above 10 Mev or so, Sec. 2B] The Data of Nuclear Reactions 23 a thin foil in which electron-positron pairs are the principal products may be used. Then a knowledge of the pair production cross section from theory or from total attenuation measurements may be used to compute the incident flux. Calorimetry is possible for strong beams of not too high energy, and special photonuclear processes may also play some part, particularly as secondary standards. Radioactive sources of gamma-rays, or gamma-ray sources which use a nuclear radiative transition like the decay of the familiar state of Be 8 at 17.6 Mev excita- tion, may be calibrated by using other particles emitted in coincidence, if the details of the disintegration scheme are known. The strong sources are usually gamma-rays produced by the stopping of electron beams by Coulomb scattering, or bremsstrahlung radiation. This radiation is of course continuous, rather than discrete, like that of the nuclear radiative decays. Its energy limit is in general well-known, because the original electron energy can be well-defined and measured. Individual quanta may be selected by using coincidence methods with the electron after the radiative act. In general, though, the whole spectrum is used, or at least that above some threshold for a given reaction, or above a broad limit set by absorbers placed in the beam to remove the softest quanta. In this case cross sections are hard to give for a definite energy. Again dif- ferential measurements may be made. Very often the cross sections for bremsstrahlen-induced reactions are given not per quantum of a definite energy, but per effective quantum in the beam. The number of effective quanta Q is denned (B16a) as the ratio Q = II n{E)E dE /2? maX) where n(E) dE is the number of quanta in the beam exposure having energy between E and E + dE and E max is the maximum energy in the beam. The rough form of the spectrum for bremsstrahlen is n(E) dE = N v dE/E; therefore in this approximation the value of Q for a brems- strahlung beam is just the number N . This is usually satisfactory if the region of interest is well above the low-energy deviations from the l/E distribution for the beam. For quantitative work the exact form of the bremsstrahlung spectrum must be taken into account and the experimental data analyzed by a numerical process indicated, for ex- ample, in (K4). Neutrons present many related problems; their detailed treatment is to be found in Part VII. 5. A Table of Yields. As orientation to the size of yields to be ex- pected in typical practical situations, Table 1 shows a number of thick- target yield values for various nuclear projectiles of widely available energies. In all cases the target is taken to be thick enough to stop the entire beam, and to be the pure element of normal isotope composition. 24 A Survey of Nuclear Reactions [Pt. VI For gamma-rays and neutrons such a target is not usually practical, but computed values are given for comparison. Only a few values are given for reactions due to other particles than the heavy charged parti- cles of central interest in this chapter. Table 7 contains additional data Yields of TABLE 1 Some Typical Nuclear Reactions (Thick-Target) Product Nuclei/10 6 Projec- Projectile Reaction tiles (beam energy in Mev) Gamma-rays AF(Y,P) 8 X 10 s (17 Mev) AF(t,w) 3 X 10 3 (17 Mev) Cu(7,n) 11 X 10 3 (17 Mev) W( Tl n) 18 X 10 3 (17 Mev) Electrons Cu 65 (e,e'n) 2 (17 Mev) Neutrons n-capture on any but He 4 10 6 S 32 (n,p) ~2 X 10 5 (5.8 Mev) Protons Cu(p,n)Zn 63 18 (6.3 Mev) Deuterons Be(d,H 3 )Be 8 230 H 3 (14 Mev) Li(d,2n)Be 7 22 (19 Mev) B(d,n)C u 1.5 (8 Mev) C(d,n)N 13 1.4 (8 Mev) Na(d,p)Na 24 190 (19 Mev) P(d,p)P 32 88 (8 Mev) ; 350 (14 Mev) K(d,p)K 42 11 (19 Mev) Ca(d,a)K 42 0.3 (19 Mev) Cu(d,p)Cu 64 330 (14 Mev) Y(d,2n)Zr 89 4.9 (14 Mev); 52 (19 Mev) Mo(d,a)Cb 90 0.4 (14 Mev) Te(d,n)I 131 1.6(8 Mev); 32 (14 Mev) Bi(d,n)Po 210 56 (14 Mev) Alpha-particles A(a,pn)K 42 17 (44 Mev) Pb(a,2n)Po 210 48 (44 Mev) Bi(a,2n)At 211 0.13 (38 Mev); 130 (44 Mev) on deuteron reactions. Reference (GO) contains still more data on charged particle beams and a very useful survey of chemical methods for preparation of carrier-free radio isotopes. C. Types of Reactions : A Guide to the Literature It is not the function of this section to present a complete survey of the thousand-odd nuclear reactions which have been studied. Even a list of reactions by types cannot be exhaustive, since reactions have been observed at high energy in which dozens of products occur. It would be possible to systematize such highly multiple reactions only by giving the relative yields of the various products. Putting aside such processes Sec. 3A] The Nuclear Model 25 for the present, most nuclear reactions involve the emission of two particles after the reaction, following the type of reaction T(i,p)Ji. A sizable number of reactions are known in which three or four definite products are emitted; then there is a continuous sequence of more complex reactions which typically occur at high energy. There is also the familiar fission reaction, in which, instead of a rather heavy nucleus emitting several light nuclear particles, the heavy target divides and two heavy fragments come off, accompanied by several neutrons and, infrequently, an alpha-particle. A systematic list of nuclear projectiles, with the products they induce, and references to original literature which describes reactions of the type so defined are presented in Ap- pendix II. The references make no attempt at completeness, nor have they been chosen for the purpose of giving the first papers on the reac- tions. They are meant to indicate a few recent and rather complete papers, especially those which contain theoretical treatment and good bibliographies. It is hoped that entries in Appendix II will assist the reader both in finding a rather general introduction to the reaction type of his interest and in getting a good start toward an exhaustive search of the literature. SECTION 3. THE NUCLEAR MODEL A. Qualitative Account of the Model We can now begin to construct a nuclear model which will have some connection with the complex reality of nuclear process and structure. In the study of nuclear reactions the comparison of the detailed implica- tions of successive models with experiment has led to a graphic but still by no means adequate model. It is this model we here seek to describe. It is instructive to compare our knowledge of the atom with that of the nucleus at its heart. We know at last that the atom is fully described as an assembly of rather well-defined electrons, held together by their electrostatic attraction to the dense central nucleus, influencing each other by their mutual electromagnetic interaction, and moving in accord with the principles of quantum mechanics (especially the exclusion principle). The nature and properties of electrons are well-known, all interactions being given in detail by Maxwell's and Lorentz' equations. Only for very few details — the hyperfine structure and isotope shift, the line shift due to electromagnetic radiation field coupling — is further investigation needed. Moreover, the well-marked periodicities and shell structure are themselves evidence that the complex mechanics of a many-body problem are susceptible in the atomic case to far-reaching 26 A Survey of Nuclear Reactions [Pt. VI simplifications in which the many-body problem is soluble by methods which begin with the idea of the independent motion of a single electron. Contrast the nucleus. Even the forces which act between pairs of nucleons are still without any general expression. In Part IV of Volume I, the two-body problem is discussed at length in an effort to find empiri- cal clues to the force laws. The nuclear forces are short-range forces; no large-scale counterpart exists for laboratory investigation. All our knowledge of nuclear forces is based on nuclear experiment itself. The nucleus is a system of many bodies, whose general mechanics is evidently more complicated than that of the equivalent atom for, although regularities exist in nuclei, they are not so striking as those evident in the periodic table of the atoms. The major regularities, important though they are (see Part IV, Volume I), cannot yet lead to confidence in a simple building-up process like that of the atomic domain. Only in the lightest nuclei, below neon, or even below helium, is more or less easy progress to be hoped for in obtaining a definite and detailed picture of nuclear matter. We shall see how the short-range and exchange character of the forces, the strong component of non-central force, the whole tightly bound character of the nuclear state of matter make difficult its detailed description. The point of view so far most successful in the discussion of reactions, though clearly itself incomplete and often misleading, involves a kind of abstraction from the detailed structure of any particular nuclear species, and an effort to see what types of behavior, what experimentally determined properties, can be used to give an account of nuclear reactions. 1. The Interaction of Nucleons. In the atom, each electron moves in a rather slowly changing force field, the sum of the nuclear attraction and the averaged repulsion of the other atomic electrons. The forces are of long-range nature, and it does not matter much if we neglect the very occasional close collisions between the other electrons and the one whose motion we are considering. We can hope to study the motion of each electron separately, taking account of the others present only in a general and smoothed-over way. The presence even of scores of electrons does not impossibly complicate the problem : we fix on the detailed mo- tion of each in turn, correcting only slightly for the specific behavior of its neighbors. This approximation, known as the Hartree approxima- tion, is excellent for the loose, smooth, open structure of the atom, gov- erned by long-range forces. But in nuclei the scheme seems entirely inappropriate. Here the forces are of a range smaller than the dimen- sions of the system. The force on each particle depends strongly on the detailed configuration of its neighbors; a small difference in the position of another nucleon can change its interaction from a decisive one to ;i, Sec. 3A] The Nuclear Model 27 negligible one. The miniature solar system has disappeared; we think instead of a miniature drop of water, a small drop of quantum-mechanical liquid. Instead of a slowly and smoothly varying mutual interaction, superimposed as a perturbation upon a stronger central force, all of which we can without much error replace by a smoothed-over time average, the interaction energy of a single nuclear particle with the rest of the nucleus is a jagged and highly fluctuating function of time. To average in space over such a function is to conceal its essential features. The continual transfer of energy from particle to particle is the rule; no well-marked center of force exists. It is misleading to try to "peel off the motion of each particle" in turn and try to correct for the average effect of its neighbors. In this picture the shell structure of loosely coupled particles has no obvious counterpart in nuclei. Yet the lightest nuclei show such proper- ties that for a long while they have been studied by the use of the Hartree approximation and related methods. Moreover, the nuclei all the way up the table exhibit regularities, in their lowest states of excitation at least, which imply a well-defined shell structure. Such properties as (1) special stability of particular values of A and Z (of which the familiar increased stability of even-even nuclei is the proto- type), (2) the angular momentum and parity of the ground state and, for many nuclei, of the lowest-lying excited states (isomeric states), and (3) rough values of the magnetic moments of ground states can be rationalized and even predicted by semi-empirical results of what is called the shell model. In the most recent and remarkably successful of the shell models (M5), the individual nucleons are pictured as moving in Hartree-Iike shells of well-defined orbital angular momentum and radial quantum number. The order and the total angular momentum of these states can be fixed by arguments based on the picture of strong spin-orbit forces, which couple the spin and orbit of each individual nucleon strongly together, so that the total angular momentum and parity can be established by the analogue of spectroscopic j-j coupling, taken with certain semi- empirical rules for deciding the order of the terms. There results the order of shells, in the familiar spectroscopic notation, shown in Table 2. There is a rapidly growing literature of applications of the shell model to the prediction of the properties of low-lying nuclear levels; it cannot be doubted that no previous effort to describe such states has been so successful, and indeed it seems demonstrated that strong spin-orbit forces must be present. How the single-particle orbit model, with its marked shells, can be reconciled with our general arguments on the unsuitability of the 28 A Survey of Nuclear Reactions [Pt. VI TABLE 2 Nucleab Shell Oeder in the j-j Coupling Model The suffix indicating the j value has been suppressed, except when the same orbital angular momentum appears in two adjoining shells; then the state of highest j is indicated as lying lower in energy. The symbol 2p, for example, means both the states 2p% and 2p^. Shell Configurations i Values Number of Nucleons in Shell Total Number (Neutron and proton shells are independent) Is lp Id, 2s 1/, 2^, lff9 /4 lgt/i, 2d, 3s, \hm Xh%, 2/, 3p, liijs A A, A A, A, A A, A, A, A, A A, A, A, A, !K A, A, A, A, A, X A 2 6 12 30 32 44 2 8 20 50 82 126 Hartree picture for the strong, short-range, and fluctuating nuclear forces is still by no means clear. But an interesting analogy due to Weisskopf seems to point out the nature of the eventual solution. He recalls the behavior of the Fermi gas of electrons, regarded as non-inter- acting, and moving in the periodic potential of a metal lattice. Here shells — the Brillouin zones — are very marked, and the neglect of the electron Coulomb interaction, certainly reasonably strong, seems to have very little effect. Qualitatively this must be ascribed to the circum- stance that electron-electron collisions can make no physical difference if all states in the momentum space are filled, as required by the exclusion principle for the Fermi gas. Such scatterings represent no change in the total system. If, however, a higher-energy electron enters the metal from without, it carries momentum adequate to excite the electrons of the Fermi gas beyond the Fermi limit; its collisions are rapid, and it loses energy very quickly to the electrons of the metal. In the same way, the low-lying states of the nuclear system may be represented well enough by the non-interacting particle picture; the expected collisions are mostly excluded. Yet for high excitations the regions of momentum space which are not filled can be reached, and the collisions become decisive. This picture fits the real nuclear situation very well. Although the stationary states of nuclei, and especially of the lighter nuclei, lie beyond the scope of this treatment, it is appropriate to men- Sec. 3A] The Nuclear Model 29 tion here the most powerful of the methods deriving from the Hartree picture, which has found great use in the discussion of just those prob- lems. The method is the Wigner theory of nuclear supermultiplets. Using an extension of the same essentially group-theoretic methods which permit the classification of the spin and orbital quantum states of the atomic system, Wigner considers a nuclear system a collection of particles interacting by forces which depend only on spatial configura- tion, not on spins or on the charge of the nucleon. Exchange forces may occur, if they are space-exchange forces only. For such interactions, certain quantum numbers can be defined, in extension of the J and S numbers, familiar in atomic multiplets, which characterize the spatial symmetry of the nuclear wave functions for any collection of nucleons. Then, by regarding the Coulomb and the spin-dependent forces as perturbations capable of splitting the highly degenerate levels of the ideal system, a more realistic picture can be obtained which has con- siderable success in the ordering of the lower-lying states of not too complex nuclei. For specific information the extensive literature may be consulted. However, it is fundamental to the picture of the nucleus which we shall chiefly employ, the picture appropriate for excited states: that energy exchange with ease between the closely packed nucleons; that the nucleus be regarded as a highly condensed state of matter, a tightly packed quantum liquid, where the relatively small number of degrees of freedom (compared to those of a drop of water), and the diffrac- tion and exchange effects to be expected from quantum considerations at such small distances be thought of as modifying the behavior of a system otherwise very like the thermodynamic system of a familiar liquid drop. We know that the radius of the sphere of nuclear matter is rather well represented by the formula R = r A y ° (23) with A the number of nucleons, and r = 1.4 X 10~ 13 cm (CIO). This is just the assertion of constant density which defines our model as a liquid or a solid, not a gas. From the simplest Fermi gas model, or what is almost the same, by taking the mean velocity of a nucleon in the nucleus to be that corre- sponding to a de Broglie wavelength equal to the mean spacing of nucleons, we find that the kinetic energy of a nucleon ought to be some tens of millions of volts. This gives a mean collision time of some 10 — 13 cm per 5 • 10 9 cm/sec, or about 10 -22 sec. For any interacting particle which takes a time longer than this to complete its collision with a bound nucleon, we have to regard the nucleus as a closed system, without a 30 A Survey of Nuclear Reactions [Pt. VI sharp separation possible between collisions with a single nucleon and collisions with the whole of the nuclear matter. Only if collisions are made by particles moving so fast, and transferring so much momentum to the struck nucleon, that the collision is complete within the charac- teristic nuclear time can we think of the nucleus as a collection of free particles. Even then the nuclear "gas" is highly degenerate; quantum levels are filled up to a rather high energy, and low momentum transfers thus are discouraged. It is then clear that, for incident particles with an energy per nuclear mass unit up to 20 or 30 Mev, the nuclear drop must be considered as a whole. Only in the higher-energy domain are we more nearly justified in thinking of a collection of instantaneously free nucleons; and even here there are important effects of the nuclear binding. Let us continue this qualitative discussion of the collision between a nucleon and the struck nucleus. If the incident nucleon is of high enough energy so that it may be localized (i.e., represented by a wave packet whose main components have wavelengths smaller than nuclear dimen- sions), we may try to follow the collision classically. The short-range nuclear forces extend about as far as the mean spacing of nucleons within the structure. A colliding particle then has very little chance to travel through the nucleus without striking the nucleons within. At very high energies (say 100 Mev or more) the mean free path for collision with the nucleons of the nucleus has dropped to some 4 X 10~ 13 cm, but, using the same rough idea of collisions with nearly free neutrons, the free path has dropped to a tenth of that value for ten-million-volt incident particles, and to only 10~ 14 cm at 1 Mev (S7). The last dis- tance is so small that the picture evidently fails, but the conclusion seems confirmed. As soon as the particle crosses the surface of the nuclear sphere, it will interact strongly with the nuclear matter. It may lose only part of its energy to the first particles it encounters, and continue on, transferring energy to the nuclear matter as it goes. Eventually its entire kinetic energy is spread in some way over the many particles of the nucleus, and a new state of quantized collective motion for all the nuclear particles exists, a nearly stationary state of what is called the compound nucleus. We shall say more of this in Section 3A2. The nuclear drop has of course an energy content, given in an under- standable form by the semi-empirical formula discussed in Section 3B4. The electromagnetic properties of the nuclear drop may be computed and measured; indeed, all the properties of the ground state can be ascribed to the model. But the details of nuclear structure cannot be obtained from such simple assumptions as underlie the semi-empirical energy formula, especially the constant density of nuclear matter. Sec. 3A] The Nuclear Model 31 Still less is to be expected from the use of this statistical model in nuclear reactions. In general, it may be said that what we here visualize is not the particular quantum level of a particular nucleus with a well-defined energy, but the average to be expected in a given region of mass number, including many similar but not identical neighboring nuclei. More formally, we shall deal with all the characteristic properties of nuclear states not one by one, but averaged over an interval of excitation energy. Only if the fluctuations in these properties are not too great from level to level, so that the average behavior represents the individual behavior of states, or if the experimental situation produces averages as the quan- tities directly observed, can we hope for satisfaction from such a statisti- cal point of view. It is in this charitable sense that the results of the theory are to be taken. We can summarize the above statements by setting out the two leading assumptions under which the results of the present theory of nuclear reactions are obtained: 1. Nuclear matter is composed of closely packed particles, strongly interacting and capable of rapid interchange of energy, like the particles of a drop of some quantum-mechanical liquid. We may call this assumption close packing. II. The results obtained from such a consideration are meant to apply not to the particular and specific properties of one well-defined state of motion, but to an average over many states of neighboring energies. Only if the property studied does not fluctuate too widely in such an interval will the conclusions be trustworthy. This we may call the statistical assumption. 2. The Compound Nucleus. In Section 3A1 we discussed the dissi- pation of the kinetic energy of some incoming particle as it passed into and merged with the closely packed particles of the nucleus. The resulting configuration we called the "compound nucleus." Let us go on with the story of such an event. After the incident particle has merged completely with the nucleus, its energy, no longer concentrated in one particle, is shared by the collective motion of all particles of the new system: the compound nucleus formed by projectile and target. Each of the nucleons will have some additional energy, but none will be likely to have all the incident energy at any one time, or even any very large fraction of it. The energy will be spread among the very many degrees of freedom exactly as the heat energy of a drop of water is kinetic energy spread among many degrees of freedom. But the compound nucleus will eventually lose its energy of excitation. Either radiation, which, as we shall see, is a relatively slow process, will finally remove the energy and thus "cool" the nucleus, or after a long time (time enough 32 A Survey of Nuclear Reactions [Pt. VI for many collisions of nucleons within the nucleus, some ten million in typical cases) the configuration will have the rather unusual property that a sizable amount of energy is concentrated on a single particle, and a particle may escape, cooling the nucleus by "evaporation." It is evident that the escaping particle will not in general be the particle that entered, and that it will have neither the energy nor the direction of the incident particle. Even if by chance a neutron, say, goes out when a neutron enters, this need not be an elastic collision. On the contrary, excitation energy will in general be left behind, to be dissipated in another way; the collision will be inelastic. Elastic collisions do not take place with formation of a compound nucleus except by the re-emission de- scribed under the special and rare circumstance that the residual nucleus is left in the single ground state only. The elastic scattering observed includes other effects by which the incident wave function is scattered coherently without the actual formation of a compound nucleus, through purely wave-mechanical processes. The formation of a compound nucleus may, it is true, take place under rather special circumstances. A gamma-quantum, with wave- length long compared to the nuclear size, may excite proton vibrations with a definite phase relationship. Or, considering the same process from a quite different point of view, the quantum may act upon a single surface proton which receives enough energy to escape without striking other nucleons on its way out. Or the proton coming out may lose energy to a single spot on the nuclear surface, "heating" the nucleus not as a whole but locally. The final reaction might occur by evaporation before the "conduction" of heat away from the hot spot to the rest of the nucleus has taken place. In all these cases — we believe them infre- quent but possible — a detailed consideration of the particular process (B17, Kl) would be required. However, we shall discuss mainly the fundamental notion of Bohr, which is that the compound nucleus is capable of disintegration in a manner independent of its method of formation. This is perhaps the third assumption of our theory, and we have seen under what circumstances it would require modification. We shall state it explicitly: III. Nuclear reactions in the domain of statistical theory take place in two separable steps: (1) the formation of a compound nucleus by combination of the incident particle and the target nucleus, and (2) the subsequent disintegration of the compound nucleus, in a manner inde- pendent of its method of formation, into the products of the reaction, the emitted particle and the residual nucleus in some one of its quantized energy states. A reaction that leads to more than two products proceeds by a continuation of this scheme: the first product comes off, and the Sec. 3A] The Nuclear Model 33 residual nucleus then acts as an excited compound nucleus, emitting another particle, and subsequently even another, and so on. Thus as many as a dozen neutrons may be "boiled off" successively from a highly excited nucleus (T4). 3. Nuclear Energy Levels and Level Widths. So far we have dis- cussed the ideas of the statistical theory from the point of view of the close-packed nucleons, an almost geometrical and quite classical argu- ment. The same conclusions can be expressed in terms of energy levels rather than in geometric terms. We have already shown that the com- pound nucleus exists for a time long compared to the mean free time for a nucleon collision in the nucleus. This means that, although the descrip- tion of the individual nucleon as free with a definite energy was very rough, the compound state can be thought of as existing with a wave function which is the approximate eigenfunction of the Hamiltonian of all the nucleons. We restrict ourselves to states which genuinely represent the compound nucleus or its disintegration products: that is, to wave functions which contain no parts corresponding to one particle at a distance from all the others, except where the distant particle is represented by an outgoing spherical wave. This serves to exclude states in which a free particle exists which could not have been emitted by the compound state. Under these conditions we can write *. = ^ n -e [E "^ ir " /2)m (24) where the energy is no longer a real eigenvalue but contains an imaginary part. This is the familiar procedure for a damped oscillator (H6). The fraction of systems in the state n will vary with time like the factor e -IW \ Thus T n /h is the reciprocal mean lifetime of the state; V n is called its width. The usual expression of the uncertainty principle AE • At c^ If is exemplified here. What does the spectrum of a nucleus look like? Simply by considering the distribution of energy among the many particles of a tightly bound system we can form some idea. We begin with the fact that the binding energy of a particle in the nucleus is only a few times smaller than the rough estimate we can make of its kinetic energy. If the particles did not interact strongly — a bad assumption — we could think of forming the excited levels of the nucleus by giving all the excitation energy to one nucleon, or dividing it among two, three, • • • , and so on. Now, since the well in which we could imagine the particles to be moving (see Fig. 3) is far from fully occupied with particles, we can find levels for each nucleon which lie quite close to its ground level. Thus the dis- tribution of a few Mev of excitation energy could be accomplished in a very great number of different ways. Most of these would correspond 34 A Survey of Nuclear Reactions [Pt. VI to the general spreading out of the excitation energy among the many nucleons; very few indeed would be the states in which all the energy was used to promote one or even a few nucleons. The coupling of the particles of course means that the levels are of mixed character. A level cannot be ascribed to a specific distribution of energy, spread among a definite set of particles. In any state, part of the energy is sometimes concentrated on one, sometimes on two, etc. Each eigenfunction could be expanded into a series of terms corresponding to the various partitions of energy. But, with so many possibilities available, the contribution E Continuum (dissociation) rz — -^ :l=5S?s=: _ ~ ~ Empty levels > Filled levels ■ y Effective potential well Fig. 3. Schematic nuclear potential well and nucleon energy levels. of a partition which corresponds to assigning all the energy to one or even two nucleons will be very small indeed. The total number of levels will evidently be extremely large. These levels, as we have said, are not completely sharp. Only the ground level is quite sharp (even it is widened imperceptibly by the possibility of beta- or alpha-decay or of spontaneous fission). As we add excitation energy, the levels become wider because of possible radiative transitions to the lower states through emission of quanta. For some sufficiently high excitation energy, the nucleus will be capable of emitting one particular particle, say a neutron. This will, of course, represent a widening of the levels. But, since so little of the wave func- tion represents a configuration with all the available energy concentrated in one neutron, the effect will be by no means large. The level width will not change markedly as a new decay process appears. The levels will simply widen progressively as more and more possibilities of decay become energetically allowable. The character of the levels will not change abruptly with their energy. Only the appearance of a small probability of emission of a particular particle will indicate the crossing of the energy threshold. As the energy of excitation increases, the total Sec. 3A] The Nuclear Model 35 width of levels rises. Moreover, the increase with energy of the number of ways the energy can be partitioned is marked. Finally, the spacing is so small and the width so large that the levels overlap and the familiar continuum has been reached. Even here fluctuations in level density and special selection rules may maintain the features of a discrete spectrum. Only when the excitation is comparable to the total binding energy — say 6 to 8 Mev per particle — does the character of the levels change qualitatively. In such a region the concentration of sufficient -t >- Continuum (dissociation) A few empty levels Atomic potential well Fig. 4. Schematic atomic potential well and electron energy levels. energy in one nucleon to allow its escape is the rule and not the exception. Under these circumstances a genuine continuum of levels exists. In Fig. 4 is shown the parallel situation in the case of an atom with many particles. From the geometrical point of view, the atom is a loose open structure. Its long-range forces mean that elastic collisions are the rule; even if an incident particle excites the atom, the energy of excitation is given to one or perhaps a few of the valence electrons. The K shell electrons have in a heavy atom about a hundred thousand volts of binding; the valence electrons, only a few. Even the L electrons are bound only by some tens of kilovolts. The possible energy levels thus are as shown in the figure: very narrow bands of close levels with great empty spaces separating the bands. Thus in general there is little chance that the energy of an incoming electron lies in a region where levels of the compound system exist. For atomic collisions these nearly stationary states are of little importance. Almost any system which has energy enough to release an electron will rapidly do so. It is inter- esting, however, that, even in a system so simple as the beryllium atom, terms have been found which give rise to broadened lines. These terms correspond to states in which the total excitation of the atom is greater 36 A Survey of Nuclear Reactions [Pt. VI than the ionization potential but is divided between two electrons. These states often have large widths and short lifetimes, because they can decay by radiationless transitions, emitting electrons. The analogy to the nuclear case is close. In molecules, of course, predissociation is a prominent effect. Although the concepts of level spacing and level width are entirely quantum-mechanical, Weisskopf has suggested a very pretty semi- classical picture of their significance. Consider a complicated many- particle system like a nucleus, but for simplicity with a large number of equally spaced levels, with energy of the nth level, E n = E + nS. We can make up a wave packet by combining a great number of energy eigenf unctions <p„(r) corresponding to the stationary states E n . The total wave function is then N / N \ ; V „ -iEn*/t I V" 1 „ -inSt/h\ t = 2-, a-nVne =1 L a n <p n e J -iBof/k Now it is evident that / 2xh\ 2 . ( 2xh' Thus we have constructed a wave function which is by no means a sta- tionary state; in it the configuration of the system repeats itself after the lapse of a time t, which we can call the recurrence time. If the level spacing is 5, the recurrence time is given by the relation t = 2irh./8. Of course, a real nucleus has levels which are not equally spaced, but a sim- ilar physical interpretation may be given to the quantity D, the mean level spacing at a given excitation energy. Closely spaced levels imply long times of recurrence, and conversely. It is also instructive to con- sider the distance traveled by an identifiable nucleon of the wave packet during the duration of one period of the motion. If the mean velocity of the nucleon is given by something like a Fermi gas model, chiefly the zero-point motion arising from the confinement within the nuclear volume, we get for the velocity v = h/MpF. Then the path length I traveled by a nucleon which had, say, just penetrated the nuclear surface from without before it found itself back in the original configuration again would be I = 2irh 2 /MDp F . Taking a level spacing of 10 ev, appropriate for the region of slow-neutron capture in heavy nuclei, the value of I ~ 10 — 6 cm, which is about a million times the nuclear diam- eter. This makes picturesque the complex motion in a compound state, and adds some confidence to our ideas about the statistical theory. On the other hand, if D = 1 Mev for the low-lying states of light nuclei, Sec. 3A] The Nuclear Model 37 the recurrence path is only a dozen diameters or so, and the picture of an orbital motion for a single particle becomes plausible once more. The level width can be given a similar interpretation. If a state has energy enough to emit a nucleon, we can imagine the packet so built as to describe the configuration existing when a given nucleon has just entered the nuclear surface. Now, after the recurrence time t, the nucleon would be back at the surface again, with the same energy, ready to leave in just the manner in which it entered. If it were actually to leave at this first recurrence, the implication would be that the lifetime of the state, h/r, was just equal to the recurrence time. Then the spac- ing and width would be of the same order, and the idea of well-defined compound state hard to justify. This is plainly one limiting case. But we must recall that at the nuclear surface there is a sharp change of potential. Within the surface, the strong nuclear forces act; outside (a distance only a couple of times 10 -13 cm outside), there are no nuclear forces at all. At this potential jump, the particle de Broglie wave suffers reflection. Suppose that the probability for penetration of the barrier is P. Then the width can be given in terms of the recurrence time: T ~ h/(r/P); and hence, in terms of the level spacing, T ~ P-D/2t. For the case in which the barrier is entirely nuclear (no Coulomb or centrifugal forces, realized by an s wave neutron), the penetrability is simply given by the familiar quantum formula P = 4k/K, where the quantities K and k are, respectively, the wave numbers for the nucleon de Broglie wave within and without the nuclear surface, and we have gone to the limit k « K. This is a very important result of our detailed theory, to be discussed later at length. Of course, in the present account we have disregarded the fact that there are alternative modes of decay, and the fact that there are constants of the motion besides energy, like angular momentum and parity, which must be conserved but which may vary from level to level. In the calculation of the appropriate D, such classes of states are to be regarded as contributing independently. In general, the statistical theory we discuss here is limited to the treatment of levels en bloc, without much hope of identifying states of a specific kind or of predicting energy levels. Since it is clear that the many well- defined observed levels in thermal neutron absorption in the heavy ele- ments (spaced as they empirically are some 10 ev, and lying 8 Mev above ground) must be not far from millionth members of some series of terms, it is not unexpected that we would make more progress, in heavy nuclei at least, by some general statistical treatment than by a method based on classifying and identifying individual terms. Individual levels can be observed at low energies in all nuclei, e.g., wherever gamma-ray spectra can be studied. In the lighter nuclei, such 38 A Survey of Nuclear Reactions [Pt. VI levels are visible up to tens of millions of volts of excitation; in heavier nuclei, not more than a million volts beyond the neutron binding energy. The most general features of observed level distribution is the steady increase of level density with excitation energy by a function which becomes the steeper as we go to heavier nuclei. A few examples of 10' _ / V N 15-11.7 hJ3 a) ft > CD 10 ; Observed averages for zero excitation A F 20-6.7 Na 24-7.2 ~ Al 28-7.9 8Mev Observed points: Z-A -excitation energy Curve models: — - ■■ ■ Semi-empirical - — — «— Fermi gas _ - — Combinatorial o 100 200 Fig. 5. Nuclear level spacing. This figure presents the nuclear level spacing as a function of the mass number A for various nuclear excitation energies. The marked curves are plots of the results of various models discussed in the text. The circles mark individual observed levels; the nucleus and its excitation are marked beside each experimental point. The three horizontal lines show the spacing observed at zero excitation, averaged over a fair number of nuclei in each region of A. The roughly satisfactory nature of the theoretical curves is shown, as well as their inade- quacy for both light nuclei and heavy nuclei, like Ni and Pb in which the influence of shell structure is marked, especially at modest excitation energy. estimated average spacing, taken from the few known levels in the various regions of interest, are given in Fig. 5. It is noteworthy that the average spacing near zero excitation decreases from the lightest to the heaviest nuclei by a small factor, perhaps by about a factor of 10, while the spacing at some 8 Mev excitation changes by five orders of magni- tude. It will, of course, not escape the reader that the application of the statistical notion of level density to the few levels of the lightest nuclei is a dubious business. Sec. 3B] The Nuclear Model 39 The width of the levels increases as their spacing decreases. We expect levels that overlap one another in particle excitation of reasonably heavy nuclei some 10 Mev above the ground state. Unfortunately the difficulty of producing closely monoenergetic beams with high kinetic energy has limited the possibilities of experimental check. The use of endothermic neutron-yielding reactions has been about the only direct test of this point. The work of Barschall with total neutron cross sec- tions at energies around \ Mev on many elements indeed serves notice that this picture is too simple (B3). The levels vary greatly in spacing and strength ; for certain nuclei the whole level structure is far too coarse- grained for the validity of our picture. This is primarily a result of the fact that the shell structure strongly modifies the binding energy of the added neutron. Excitation by neutron capture with low kinetic energy may excite some nuclei (those with closed shells) by only a fraction of the usual 8 Mev. At the lower excitations, the level density is much reduced. Other reasons may also be involved. B. Calculation of Level Densities Again disregarding certain details of structure, we may proceed to calculate mean level densities for some statistical models of the nucleus. We shall present two of the most instructive in some detail, giving only a sketchy account of some other schemes which have been proposed. 1. Combinatorial Model. We have already discussed qualitatively the simple model for which we here calculate the mean level density. It is based upon the idea of a set of independent particles among which we divide the energy of excitation. Suppose that each of the N particles has an identical energy spectrum, which we shall take to be of the sim- plest possible form: a set of equidistant levels of uniform spacing 8. Then the excitation energy W can be written as W = £ m& i where the n t are integers showing which level of its own spectrum each particle occupies. A given energy W can then be realized in as many different ways as there are ways to sum integers to a given integer total. The states thus formed would be highly degenerate, the degree of de- generacy being equal to the number of ways in which the fixed sum can be realized. If we now take into account the interaction of the particles, the degeneracy will be split, and each level will be split into as many levels of the whole system as the value of its degree of degeneracy. In this way the level density can be calculated. The result is based upon the asymptotic formula of Hardy and Ramanujan (H4) for the number 40 A Survey of Nuclear Reactions [Pt. VI of different ways, p(n), to form a given number n by any of the possible sums of smaller integers: p( n )=_J_ „*<*»>* 4(3) ^n If we take as a value for the unit level spacing 5 the average observed separation between the lowest levels of heavy nuclei, 2 • 10 5 ev, we get, for the degree of degeneracy expected for a total excitation energy W of 8 Mev, Q- pi—) = p(40) = 4-10 4 which gives an average level separation for the whole excited nucleus of about 5 ev, which very roughly corresponds to the density of level separation estimated from slow-neutron collisions. The level separation, 4.(3)^We~' iH ' w/s)>i p(W/sy depends exponentially on the excitation energy; this is characteristic of all nuclear models and is displayed even by this very simple one. 2. Gas-Like Models. In this model the nucleus is visualized, in the first approximation, as consisting of gas — a set of non-interacting par- ticles — held in a spherical box of nuclear size. The two Fermi gases present (neutrons and protons) occupy all the lowest levels in the box. As the nucleus is excited — "heated" — some of the higher levels are occupied with increasing excitation. Bethe has given a careful statistical calculation for this model, following the usual methods of statistical mechanics (B13). He has used a statistical ensemble with energy, particle number, and angular momentum taken as given constants. The calculation is of course quite parallel to the calculation of the entropy of the system, since entropy is proportional to the logarithm of the number of states of the system. Somewhat greater care is required for the computation of the number of states than for its logarithm. If W is the energy of excitation and A the number of nucleons, he obtains, for the mean spacing D (in ev) of nuclear levels of some definite angular momentum /, D = 5 .ltf . (^}\ e-Uw/2.1)" (25) In this formula we have assumed the usual r = 1.5 • 10" is in Mev. Sec. 3B] The Nuclear Model 41 A similar but more sophisticated model was proposed by Bardeen (B2). He solved essentially the same problem, but he made use of the proper- ties of the exchange forces in nuclei. It has been shown that the two- body problem in a square well with exchange forces could be reduced to the one-body problem except that the potential acting on one particle now depends on the wave number of the interacting particle. It turns out that the total energy of the equivalent particle motion increases more strongly with the wave number than the kinetic energy alone. This effectively increases the phase volume available for levels, reduces the number of individual particle states, and therefore much more strongly reduces the density of levels of the whole system. The effect is quite analogous to the effect of the correlations of electrons which obey the exclusion principle in changing the interaction energy of elec- trons in a metallic lattice. The model of Bardeen is often referred to as the free-particle model with correlations. The energy dependence of the result is given by Eq. (25) with (ATT/2. 1) replaced by (ATT/4). 3. The Liquid Drop. The idea of regarding the nucleus as a classical drop, with volume energy and surface tension, is by now a familiar one. On this basis the normal modes of vibration of a liquid drop may be computed: there are surface vibrations and transverse and longitudinal volume waves. The surface energy and the compressibility of the nuclear matter may be estimated from the semi-empirical energy content given in Section 10. Then the vibrations of each normal mode are thought of as quantized, and energy distributed among them according to Planck's distribution law. The nuclear matter is so "cool" that equipartition is far from achieved; the excitation energy is not very large compared to the zero-point energy. From the treatment of Bethe, numerical results can be obtained (B13). Here again the energy de- pendence is given roughly by the formula Z)~exp (-cTT M ), TT small D ~ exp (-c'TT M ), TT large (26) It is interesting that in this treatment one can show that only a few normal modes of surface vibration are present for heavy nuclei in the region of 1-Mev excitation or lower. The possibility of rotation of the drop, associated of course with its angular momentum and the multipole moments of its low-lying states, has been considered. Though the idea is a satisfying one, it can be said that the drop model does not account quantitatively for the low-lying levels of high angular momentum which seem to be responsible for nuclear isomerism. On the contrary, direct calculation of the spacing of rotational levels gives for their energy, for a lead nucleus say, only a few kilovolts. Such fine structure has not yet 42 A Survey of Nuclear Reactions [Pt. VI been observed. It has been pointed out that particles whose wave func- tions must satisfy symmetry properties may, unless very rigidly bound, be unable to rotate as a rigid body rotates because some rotational states of motion are not allowed. Indeed, it has been shown (T3) that only for high angular momenta, of the order of 50 units, will nuclear rotational levels have the order normal for a rigid body or a classical drop rotating as a whole. At such angular momenta, the excitation energy is some Mev, and vibrational motion is of course important. The suggestion that a drop or a solid be used to represent the lowest states of nuclei has not proved very satisfactory; as a matter of fact, the magnetic moments of ground states are much better represented by the idea of an odd particle rotating about a non-rotating core containing all but the odd nucleon (S4). Again, we may expect better results for the drop model in highly excited states than near ground. A formulation of the structure of nuclei, where individual level properties are decisive, is still beyond the scope of the present models, although for the lowest states the individual-particle shell model in j-j coupling form has had remarkable success. 4. Thermodynamic and Semi-Empirical Results. The general behav- ior of the level density as a function of energy is similar in all the models discussed. The details are not correct, and the constants deduced com- pare indifferently with experiment. It seems reasonable then to compute a general form of the level density as a function of energy from the most general statistical or "thermodynamical" point of view, and to adjust the constants until we get reasonable agreement with the not very well- known level spacings observed. Let us begin with the fundamental statistical-mechanical relation for the sum over states (B13) z=Z. -Ei/r where the values E t are the energies of the states of the whole system, and r is the parameter which we can identify with the usual kT, the temperature measured in units of energy. Since we are dealing with dense levels, we can rewrite the sum as an integral : Z=Jp(E)e~ }s ^dE (27) But let us introduce the function A, the Helmholtz free energy of thermo- dynamics, by the relation: Z = e~ AlT . Now, if the system has a large number of degrees of freedom and the distribution is one of equilibrium, we expect that the contributions to the integral in (27) come mainly Sec. 3B] The Nuclear Model 43 from the region near the maximum, which we call W, the mean excita- tion energy of the nuclear system. We then rewrite the expression as Z = Ce ln '>- ElT dE /a(lnp) a'lnp (ig-WO 2 i -(g-WQ e -al- (E_,,0+ -8^ 2 + ; rf# Expanding the integrand about its maximum by the familiar saddle- point method, we obtain w , T /a 2 (ln P -£?/r)| 1* Z = P (W)e-^/' • -2x/ * \ = e (28) L / ah \w-i Now let us introduce the familiar thermodynamic function, the entropy S, by the relation A = W - tS (29a) We expect this function to be proportional to the logarithm of the den- sity of states. From (29a) we get the elementary definition of entropy : dS 1 = - (29b) dW r and we can use this in (28) with d(lnp)/dE \ W = 1/t, to get e S(W) o(W) = T{2wdW/drY A We then have, for the general connection between density of states p = 1/D and the thermal properties of the system, J." dW'/r P (W) = v (30) K t(2tt dW/dr) 1A To determine the level density for any system, then, we need to find the entropy of the system, which we can always do by integrating (29b), if we know only the expression for the heat content as a function of temperature, that is, the relation between excitation energy and the temperature parameter t(W). Next, the procedure is either to adopt a model, and so work out the heat content as we have indicated in the previous sections, or to make some general assumptions for the temperature dependence of the heat content, and then fit the constants to the data. We can proceed in a general way by using thermodynamic arguments. We expect the heat capacity for nearly any system to be a steadily 44 A Survey of Nuclear Reactions [Pt. VI increasing function of the temperature. If the temperature of a system is so high that for all the degrees of freedom the energy r is large com- pared to the quantum energy ho> , the classical limit of equipartition is reached, and the heat capacity is constant. But the nucleus, as we have said before, is generally rather "cold." The simple model of the Fermi gas of A nucleons in volume -f ttR 3 yields the result (M4) that the "gas" is degenerate if the excitation energy is small compared to Ah 2 /2MR 2 ~ 8A H Mev. Then heavy nuclei are highly degenerate below excitation energies of some 50 Mev. The interaction of nucleons does not affect this conclusion, just as the electron gas in metals is degenerate in spite of the electron-electron forces. Now we can write the heat content as a function of temperature in the power series: W = c + c lT + c 2 t 2 -j From the general quantum ideas of the third law of thermodynamics, we expect the specific heat to vanish as T — > 0. Thus the expansion of W near zero temperature, i.e., in the strongly degenerate region, ought to begin with at least a quadratic term, with all higher powers negligible. The familiar Debye specific heat for metals x actually vanishes at low temperatures with an even higher power of the temperature, since the heat content goes to zero like 2 14 . If we assume then any simple power law dependence, W = cr n , we can integrate dS _dS dr 1 dW~ drdW~ t letting n S = (n-1) With this expression, we obtain for p(W) the value > m " ^ (r^)"' 2 «"■' (<dhj ' c "" w """) For the Fermi gas, the exponent contains 1 — K, for the liquid drop the power W* to W H at low and high temperatures respectively. It seems appropriate to use the very simple form P = Cexp(aJF)^ (31 a ) If we do this, and try to make a rough fit to the data, we obtain (following Weisskopf) (W5) for nuclei heavier than A = 60 a result which is approx- 1 See (M4), p. 253, for formula; p. 248 for derivation. Sec. 4] Nuclear Level Widths 45 imated by the simple form, for nuclei of odd or even Z and N — A — Z, 1 12 ~ ^odd-odd ~ (-"even-odd ~ ^ odd-even = *b even-even = T~. T (JVIev) 2 (.4-40) a = 3.4(A - 40)^(Mev)- 1 (31b) We present in Fig. 5 a few selected data on observed level spacing, together with the values calculated from the three nuclear models described. There is some inconsistency in what the figures represent, for the experimental data from slow-neutron capture, as well as the free-particle model calculation, refer to levels of a definite angular mo- mentum (or within ±Jh from the ground state). The thermodynamic calculation certainly includes all levels. Here there is a chance to add some more definite assertion to the general statistical notions; all we can do at present is regard all levels as more or less uniformly distributed in regard to most not very large values of angular momenta. This vague statement is typical of our statistical theory, and it may account for discrepancies observed in individual cases where specific angular momentum properties may be involved. The least we can do to recog- nize the great importance of nuclear energy shells is to use differing constants for odd- or even-Z nuclei. SECTION 4. NUCLEAR LEVEL WIDTHS We have already mentioned the concept of level width, which is defined as r„ in the expression for the approximately time-independent (because long-lived) quasi-stationary state wave function for the compound nucleus, From this form for the decaying amplitude of the wave function the width can be related to the mean lifetime in the usual way: r n = h/i-„. This expression naturally refers to the change in amplitude with time of the compound state regardless of the products of its decay; it is the total width, and r n is the total lifetime. Naturally, the compound state may in general decay in more than one way; a single set of reactants may have many alternative sets of products. We may generalize the notion of width to include the idea of partial width by writing T n = 2 rVa(w), where the r ta are the partial widths, which we index here, ia somewhat redundantly, with two indices: i, which designates the kind of particle emitted, as neutron, gamma, proton, etc.; and the index a, which gives the state of the residual nucleus. It is evident that the same 46 A Survey of Nuclear Reactions [Pt. VI particle can be emitted with different properties, as different energy or angular momentum, leaving behind generally, though not necessarily, a different state of the residual nucleus. The argument of the function Ti a (ri) refers to the state n of the compound nucleus whose width is here written. Clearly the partial width for the emission of, say, a neutron to any possible state of the residual nucleus might be the experimentally useful quantity, and it would involve the sum ^ r neutpa (n) over all a.neut accessible states which can be left behind after a neutron is emitted. To speak somewhat academically, r„ is strictly zero for almost no states of nuclei. Even the ground state is not wholly stable; in radio- active nuclei it may decay by beta- or alpha-emission, and in nuclei not observed to be radioactive we know that in general a state of less energy content can be reached either by a process like spontaneous fission or, if the nucleus is a rather light one, by combination with a neighboring nucleus of the matter present. Such lifetimes are long, perhaps enor- mously long, compared to actual nuclear reaction times, under terrestrial conditions at least. It -is conventional therefore to disregard such forms of radioactive decay, and especially beta-decay, and to regard the levels as widened only by the possibility of radiation or by the emission of a heavy particle. This implies that, while I^ may be different from zero, its study belongs properly to radioactivity theory, and not to the dynam- ics of nuclear reactions. The levels of a typical nucleus then can decay only by gamma-emission, which slowly widens the states as their excita- tion increases until the excitation energy reaches a value which makes heavy-particle emission energetically possible. For nuclei whose ground states are empirically alpha-radioactive, of course, this condition is ful- filled for every level, but for most nuclei from the middle to the top of the periodic table such a value is reached only at the binding energy of a proton or neutron. The binding energy lies near 8 Mev for the entire middle of the table; for the lightest nuclei, up to neon, say, it is widely varying; for the nuclei from gold or lead and up, it has fallen to 5 or 6 Mev (W3). Up to a level of the compound nucleus, then, which has an energy content at least 5 to 8 Mev above the ground state, only gamma-radiation will introduce a finite width; beyond it, first the emission of one particle, then of another, and finally of even more kinds and a greater number of particles will widen each level. Moreover, not only do the values of T(n) increase because there are more possible I\-, but in addition more states are available in the residual nucleus, so that a takes on more possible values. The widths themselves increase slowly, as we shall see, with the kinetic energy of the emitted particles and the energy of the gamma-rays. Sec. 4A] Nuclear Level Widths 47 A. Level Widths and Reaction Cross Sections : Statistical Relations We have already indicated that the fundamental idea underlying the present statistical treatment of nuclear reactions is the separation of the process into two independent steps: the formation of the compound nucleus, and its subsequent independent decay. Upon the basis of this assumption there can be derived a useful relation between level widths and cross sections. The method of the derivation is the familiar statis- tical-mechanical one of detailed balancing (W5). Let us consider an equilibrium mixture of three kinds of systems: the compound nucleus we are studying, and its two decay products, the "emitted particle" and the residual nucleus. The equilibrium reactions are the decay of the compound, and its formation by recombination of the two decay products. We may write C* *± p + R a We do not consider every possible state of the residual nucleus R and the particle p which might combine to form the given state of the com- pound nucleus, but only particles of such energy, angular momentum, etc., that they can combine with a residual nucleus ~R a in a definite state a to produce a particular state of the compound nucleus C*. It is clear that we must be sure of a random statistical ensemble: we must be cer- tain that special phase relations do not exist between the wave functions of the decay products and the compound nucleus which emits them or is formed from their combination. To insure this, and to free ourselves from the possible special character of individual states of the compound nucleus, we have to regard the levels of the entire system as very closely spaced, and be sure that the energy interval is large enough to include many levels of the system. Our widths and cross sections are then aver- ages over this small interval of energy, small compared to the scale of variation we- are interested in, but large enough to include many levels. All phase relations are then lost in the averaging. This is .the usual procedure required to define a quantum-statistical ensemble. Now we apply the almost revered formula of the time-dependent perturbation theory, which gives the transition probability per unit time for the transition from the almost stationary state (A) of a system to a continuum of states B at the same energy. This is 2tt . ., w A -,B •= —\Mab\pb® h where [ Mab \ 2 is the square of the matrix element of the perturbing operator between the initial and final states involved (the final state is 48 A Survey of Nuclear Reactions [Pt. VI chosen as any one of the continuum having the correct energy), and pb is the density of states of the final system, per unit energy in the volume within which the wave functions are normalized. Now the inverse process is given by the same formula written for w B ^ a- But, since the operator is certainly Hermitean, | Mab \ 2 = \ M B a | 2 - We then obtain the expression for detailed balancing in its general form WA ^B-PA = W B ~,A'PB (32) The use of perturbation theory conceals the generality of the result; any order of the perturbation will give the same dependence on the absolute square of a Hermitian matrix element, so that the result is quite general. Applying it to the situation described above, with a compound nucleus of excitation energy W decaying to two reaction products, we may write Tpa(W) 4TP 2 dp — p c (W) = Vffpa-ga. — — • — - n (2irhy dW Here we have used the relation between width and lifetime, w = 1/V = r/h, and found p B by using the density of states for a free par- ticle with mass m p , momentum p, velocity v, and kinetic energy E p , 47rp 2 /(2irh) 3 , and converting the density in momentum space to the density in energy by the factor dp/dW. Note also the important relation w = v<x for any collision between two particles of relative velocity v, in normalizing volume. The factor gr R is the statistical weight (degree of degeneracy) of the residual nuclear state. We may finally write T pa (W ) PC (W) =^-~ with X = — (33) 2jt 7rX m p v This formula tells us that the mean level width, measured in units of the level spacing, is just equal to the cross section for the inverse reaction, a pa , measured in units of the maximum cross section for absorption of particles of the given kinetic energy, apart from small statistical weight factors. It is interesting to recall that the maximum absorption cross section for s waves is just 4irX 2 , and thus the width for s wave emission is at most 2g R /w times the level spacing. The use of the detailed balancing formula for connecting lifetimes to cross sections as demonstrated here is only one of its applications. By considering an over-all reaction i + T — » p + R, two cross sections may be similarly connected. In general, viTPi 2 = <r P RP P 2 where p; and p p are respectively the initial and final relative momenta. Sec. 4B] Nuclear Level Widths 49 In (33) we could have defined T pa and p c somewhat freely. They might contain averages over all levels in a narrow band of energy, or they might be restricted to levels of a single class, e.g., only those for which the angular momentum involved could in fact be given to the particle p. Since the level densities are not accurately known, such distinctions are ordinarily not carefully made. B. Calculation of Level Widths The assumption of the independent decay of the compound nucleus can be exploited in detail for the calculation of level widths. With this idea we combine the evident fact that the forces on nuclear particles which operate at large distances from the nucleus — distances large com- pared to the range of nuclear forces — are, we believe, completely known. These forces are, of course, (1) the very important Coulomb force on all charged particles, and (2) the effect of angular momentum, the centrifugal force associated with non-vanishing angular momentum. Most of the success of the statistical theory really comes from the fact that nuclear reactions are greatly affected by these essentially non- nuclear forces, and separation and calculation of their consequences is itself a considerable step toward explanation of the most evident phe- nomena of the reactions. We shall show that under suitable conditions the partial width for emission of a definite particle with given orbital angular momentum I and energy E P) leaving behind a particular state of the residual nucleus, can be written as the product of three factors: T p ia(W n ) = ky(n)P pl (Ep) (34a) where k is mv/h. for particle p, y(n) is a factor which depends only upon the state of the compound nucleus, particularly its excitation energy W n , and P P i is the usual coefficient of penetrability of whatever barrier the particle encounters from electrostatic or centrifugal forces. Let us begin by writing the total width of a given state of the com- pound nucleus, which we shall index by T(n), as a sum over the partial widths, one for emission of each kind of particle indexed by i, with angu- lar momentum Z, leaving the residual nucleus in state a. [Compare (33) and (34a, b).] This implies that the wave equation H^ n = W n ^ n for the whole system has been solved by the wave function V n = 2p a fo, where by i we index all possible emitted particles, for which ^ } - is the wave function outside the nucleus, and <p a is the wave function of the corre- sponding residual nucleus in state a. We obtain a set of discrete but widened levels by imposing this condition on the solution, with the specification that the function \pi contain only outgoing and no incoming 50 A Survey of Nuclear Reactions [Pt. VI spherical waves. Below the energy for which some particle can be emitted, the function ^; will correspond to a bound state for that particle; above the threshold, to an outgoing spherical wave. It may be noted that, just as the presence of non-zero T(n) means that the states of the compound nucleus are broadened, and that their amplitude decays, so the complex W„ = E n — iT n /2 results in a complex wave number k p for the outgoing particle. The amplitude of the outgoing particle wave function is greater at large distances, since the particles represented as farthest from the nucleus were emitted earliest, when the compound nucleus had its largest amplitude. The asymptotic expression for the radial part of an outgoing wave e ikr /r will be damped by the presence of an imaginary term in k. Below the threshold for emission, this is the only term. Above the threshold, we can neglect this imaginary part so long as the width T n is small compared to the kinetic energy of the emitted particle. With this in mind, we write Yi m (6,<p)R pl (r p ) *p = 2-, l,m r v where R p i/r p is the radial part of the wave function of particle p, and Y e m is the normalized spherical harmonic. 1. Barrier Penetration. The value of a partial width T p i{n) is just h/rj,;, and hence just h times the number of particles emitted per second from one compound nucleus. We normalize R p i by the condition that I | 'Apa \ 2 dV — 1, where the integral is extended over the spatial i/nucl region "within the nucleus," say out to a fixed distance R. This implies that T P i is small, so that we can neglect the more remote part of the wave function of the compound state. This is closely related to our general assumptions. Then the width is just h times the outgoing flux of particles integrated over a distant sphere of radius r x . We obtain r h 2 T p i(W„) = hrj- v x \ f pl {r x ) | 2 <Kl = — • k\ R P i | 2 (34b) J m p We have required that very far away, beyond a radius r x , the radial dependence of R p i be simply e z ^ kr+s) . This means that we have chosen r x so large that in that region the kinetic energy h. 2 k 2 /2m is very large compared to the electrostatic or centrifugal barriers. From this point on, the flux of outgoing particles falls off only geometrically, with 1/r 2 . Hence the width becomes independent of r. But, in the region between the radius R which marks the boundary of the compound nucleus, and r w , which is the force-free region, the behavior of the wave function is Sec. 4B] Nuclear Level Widths 51 fully known. It is attenuated by the requirement of penetration through a potential barrier, which may be zero for neutrons of zero orbital angu- lar momentum, but would be very considerable for alpha-particles of any orbital angular momentum if the kinetic energy were not high com- pared to the Coulomb energy, and would be appreciable even for neutral particles of non-zero angular momentum. We shall write R p i(r x ) in terms of the value of R P i{R) at the nuclear surface R. The quantity P p i, which we call the penetrability of the barrier, we define as the fraction of the initial intensity of particles, of the type indexed by p and angular momentum I, with kinetic energy E, which penetrates to a field-free region. We can write the Schrodinger equation in the field- containing region for the radial variation only: \ d 2 2m P/ 1(1+1)1 If the solution F pt is that which corresponds to a unit outgoing wave, with asymptotic value e +i( - kr+i) , then evidently _ external intensity 1 intensity at nuclear surface | F v i(R) \ 2 For uncharged particles the solutions F pi are well known to be the half-integer-order Bessel functions. The boundary conditions are satis- fied by the so-called Hankel function of the first kind, M?r H l+ y 2 w (kr) A few penetrabilities for uncharged particles of various angular momenta can be given: Pnfl = 1 Pn.l = — — "T P, x 2 ~ 4 n,2 1 + x 2 "■' x * + 3x 2 + 9 x 6 n ' 3 ~ ?+te* + 4& a + 225 (36a) with x = kR. And, for general I values, if the wavelength is long or I large, i.e., if x — kR <JC I = x» n ' 1 ~ (21 - l)\2l ~ 3) 2 (2l - 5) 2 • • • l 2 (36b) A table of such functions has been published (Ul). The calculation of Coulomb barriers is much more difficult, because of the well-known slow convergence of the asymptotic expressions for 52 A Survey of Nuclear Reactions [Pt. VI Coulomb wave functions. There are two approaches : an exact calcula- tion can be made by using the published tables of Coulomb wave func- tions (Y5). Such a calculation has been made and graphs published for exact penetrabilities for protons for various angular momenta emitted from the lightest nuclei, from Li to F (C5). The other method is the familiar approximate solution for the wave equation in the classical limit. This is the W.K.B. method, useful under conditions in which the distance traversed by the particle while its wavelength changes appre- ciably is many wavelengths. If the usual graph of the energy barrier is recalled, it will be seen that the approximation is good only when the energy of the particle does not come near the top of the barrier. This means that the W.K.B. method is useful either in the trivial case far above the barrier when the penetrability is evidently unity, or in the very useful case of low penetrability. In the transition region, for P of the order of unity, the W.K.B. approximation is not reliable. The relevant formulas will be given here, though they are common in the literature. The Coulomb penetrability from the W.K.B. approximation can be written Ppi(E) = ^— - 1) exp (-2C,) (37a) Here Bi is the barrier height for a particle of charge ze, reduced mass m — nipM/nip + M, and angular momentum I; Ze is the nuclear charge and R the nuclear radius. The quantity Ci is the familiar phase integral /2m\ H r r > Ci = [ — ) J (V-E)^dr (37b) where the potential energy, including Coulomb and centrifugal terms, is zZe 2 1(1 + l)h 2 V(r) = + ~- and V(r e ) - E = 0; B t = V(R) for any I r 2mr The value for Ci is complicated: Ci 1 /tt .1-2* 1 Ar 1 - 2» \ — = — I — f- arc sin — 1 — (1 + y — x) C A 2(x)^\2 (1 + 431/)*/ H /2zZe 2 RmV A 2(x) H \2 (1 + Axy)' (1+2(2/)%* +(1 + 2/ -x)*]) + (y) log ^Ti^ (37c) with y = 1(1+ l)/(2zZe 2 Rm/h 2 ), and x = E/B t . A plot of Ci has been published for a sizable range of x and y; when used with the other Sec. 4B] Nuclear Level Widths 53 graphs in Appendix I, it makes the necessarily long computations as convenient as can be expected. It is plain from formulas (37) that the orbital angular momentum of the partial wave concerned plays a large part in determining the ampli- tude at the nuclear surface. High orbital momenta mean higher centrifu- gal barriers, and the penetrability through the centrifugal barrier may be small for high I, even when the simple Coulomb barrier has been sur- mounted. It is useful to plot a few barrier heights to show where this approximation may be expected to be satisfactory. We assume the usual formula R = r A* with r = 1.5 X 10 -13 cm. TABLE 3 Barrier Heights (in Mev) tor Various Particles and Angular Momenta f Nucleus Angular Particle Mo- Ne 10 20 Ca 2 „ 40 Zn 30 66 Sn 60 112 Yb 70 m Th 90 232 Emitted menta V 3.5 5.6 7.1 10.0 12.1 14.1 1 6.1 7.2 8.3 10.8 12.7 14.5 2 11.1 10.4 10.6 12.4 13.8 15.5 3 18.7 15.2 14.0 14.8 15.6 17.0 d 3.5 5.6 7.1 10.0 12.1 14.1 1 4.8 6.4 7.7 10.4 12.4 14.3 2 7.3 8.0 8.8 11.1 13.0 14.8 3 11.1 10.4 10.6 12.3 13.9 15.5 a 7.1 11.2 14.3 20.0 24.1 28.1 1 7.7 11.6 14.6 20.1 24.3 28.2 2 9.0 12.4 15.2 20.5 24.5 28.5 3 10.9 13.6 16.0 21.1 25.0 28.8 t These barrier heights were computed by using R = 1.5 X 10 -13 A 1 ^ cm. Now we can collect the results of the last paragraphs. Using (34b) and (35), we can write h 2 T P i(W) = — -*-| R p i(R) \ 2 -P p i{E) (38) We identify this expression with our formula (34a), if we set for the factor G h 2 m v R P i(R) (39) 54 A Survey of Nuclear Reactions [Pt. VI It is evident that the effect of the external barrier is contained wholly in Ppi) we can call the product ky p i the "width without barrier." The argument of the function R p i contains explicitly the nuclear radius, and the function represents the motion of the emitted particle within the . reach of the specific nuclear forces. It is clearly a function of the internal state of the compound nucleus, as we had expected for y. Strictly speaking, it will also depend on the wave function and its derivatives outside the nucleus, on the kind of particle, and so on, for the function R P i must be a continuous solution of the wave equation. It is the point of the statistical treatment to ignore the dependence of 7 on the external part of the wave function. Especially if we remember that the nuclear forces are strong, that the radius R is rather well-defined though not sharply so, and that we should try to apply the theory to the average behavior of levels and not to specific ones, will we accept this assertion. A more formal argument has been given, but it cannot be made air- tight because of the very nature of the theory. It seems more satis- factory to accept this simplification in the spirit of the main Bohr assumption: that the compound nucleus decays in ways independent of its mode of formation and of the details of single levels. SECTION 5. THE COURSE OF NUCLEAR REACTIONS A. The Steps of the Reaction We can now proceed to the discussion of the nuclear reaction as a whole. The process of formation of a compound nucleus and its subse- quent disintegration can be described by a reaction cross section written this way: „ ._. t _ ,,„. J <j iv = Si(E)-H-D p (40) Here Si is a cross section for reaching the nuclear matter, a process sup- posed to depend only on extra-nuclear forces; £ is a number of the order of unity which is called the sticking probability. It is roughly the probability that the incident particle will enter to form the compound nucleus once it has reached the nuclear radius. The factor f will carry our uncertainty about the validity of the statistical assumptions; it is in this factor that we can throw all the features about the phase relation- ship of the possible decay products which form part of the wave function of the compound state. If the nucleus is excited only at a "hot spot," where the lifetime of the whole compound nucleus is long compared to the time of possible emission of particles from the locally excited region of only a few nucleons, we expect the sticking probability to decrease. In general, the more nearly independent of energy and exact reaction % is, the more satisfactory is the statistical model. Sec. 5B] The Course of Nuclear Reactions 55 We have written as a separate factor, again in the spirit of our ruling assumptions, the probability D p of disintegration of the compound nucleus, once formed, into the particle p. This can, of course, be written in terms of the various partial widths: Dp = tj< (41) where the denominator contains a term for all the possible products of the disintegration of the compound state. The possibility of alternative reactions, and their relationships as the incident particle type and energy vary, is evidently expressed by the changing magnitudes of the various T's. B. The Contact Cross Section 1. Elastic Scattering and Diffraction Effects. The value of <j pl will, of course, not really be a smooth function of energy. It will reflect, through a variation in D p , the presence of resonances where they are well defined. For energies high enough so that the classical idea of the collision can be applied, but not so high that the collision times for bound nucleons are long compared to the time of interaction of the incoming particle, the simplest picture of the compound nucleus would lead us to think of £ as approaching unity: every particle that comes to the surface of the nucleus — as well as that can be defined — will stick to form the compound state. We should like to compute the factor Si by referring as completely as possible to the region outside the nuclear volume. This is of course not rigorously possible. The incoming wave which represents the inci- dent particle will be modified by the presence of the nucleus in many ways. Those scattered components which are not coherent with the incoming wave, but which represent inelastic scattering or the occur- rence of a genuine nuclear reaction, can all be referred to events within the compound nucleus. There are scattered coherent waves which represent the actual formation of a compound nucleus, with the subse- quent chance emission of a particle of the original energy and type. Much more important coherent scattered components, however, come from the effects of the extranuclear forces, from the diffraction by the geometrical surface of the nucleus, and from the "reflection" by the nuclear potential. These cannot be distinguished physically in any way from the elastic scattering following compound nucleus formation. The inverse process alone provides a means for evaluating the probability of such elastic scattering by compound nucleus formation, and thus a precise definition for <S,. 56 A Survey of Nuclear Reactions [Pt. VI Let us write the usual expression 1 for the wave function of an incident plane wave of particles of wave number k = mv/h., where m is the mass and v the velocity of the incoming beam in the center-of-mass rest system. This expression is a sum of partial waves written in spherical coordinates, each corresponding to a given value of the orbital angular momentum. We normalize to an incoming flux of unity by writing * = j-u **" = —^u Z (21 + l)^-i l -J l+ y 2 {kr)Y l \e) (42a) (v) /2 (2krv) /2 1 =0 Here Ji+y 2 is the Bessel function, and Y° the normalized spherical harmonic. At great distances from the scattering center we obtain f _* (lY . 1 J2 (21 + l)X.il.( e -«*r+lW2) _ e -N(*r+I,/2) )Fj (42b) \tv kr o This is in the absence of any effect of the scatterer. But the scatterer may change in general the phase and amplitude of the outgoing part of the wave at large distances. Suppose that the relative phase and amplitude of the outgoing wave after scattering is given by the complex number 77. Clearly | 17 | is at most 1. Then the wave function in the presence of the scatterer is no longer given by (42b) but by a sum of incoming and outgoing waves proportional to e~ lkr and e +lkr respec- tively. The incoming portion is fcn = S*in (,) , *in W - (-)" ^ (2* + 1) *6-«*'+"' 2 > F,<°> (43a) \v/ kr and the outgoing ^out = 2>/w W , *out (I) = ( T X(- (21 + l)*(+„)e+«*'+"'»r I < , » \v/ kr (43b) For a particular value of I the number of particles absorbed per second is just the difference between incoming and outgoing flux evaluated over a large sphere: r 2 f[v\ hn w ! 2 - v\ +oJ l) | 2 ] dSl = ^ (21 + 1)(1 - | „ | 2 ) (44a) Since we normalized the incoming wave to unit flux in the beam, the cross section for absorption is given by •Tabs = 2ff ab . (,) <7 ab s W = J (21 + 1)(1 - I i, | 2 ) (44b) 1 This whole treatment follows closely that of (F7). Sec. 5B] The Course of Nuclear Reactions 57 The elastically scattered wave has to be added to a plane wave in order to get the actual perturbed wave ^ in + ^ out . This is simply lAin + i^out — e* 70) H - Again the flux in this wave integrated over a large sphere gives the elastic scattering cross section: a ei = W° ffel « = ^ (21 + 1)|(1 + v )\ 2 (44c) It is valuable to consider the relations between these waves. The total cross section <r tot = 2<r tot w can be written: <rtot W = <r e l (0 + <r abs W =—(21+ 1)[1 + jfcfo)] (45) /c There is a well-known and interesting result which follows from this formula. Consider the cross section of an obstacle in the classical limit, with the radius (R) of the scatterer very large compared to X = 1/k. We expect the classical absorption cross section, summed of course over all values of I to obtain the total cross section, to be wR 2 if the object is "black" i.e., if it absorbs every particle whose trajectory strikes it. But in this case v = 0, and the total cross section <r tot = 2wR 2 . Where does the additional ttR 2 of elastic cross section arise? This is the familiar "shadow scattering," now observed repeatedly for neutrons of 90 Mev and less on heavy nuclei (CIO). The black sphere clearly casts a shadow m the beam. This shadow must, on the wave picture, as in the familiar Kirchhoff treatment of physical optics, be produced as the result of interference between the incident wave and a scattered wave. The total intensity of scattering, to produce a shadow the size of the object, evidently corresponds to a cross section just equal to the geometrical one, tR 2 . Thus there must be an elastic cross section of irR 2 . This can be observed because, especially in the nuclear case, the shadow does not extend to infinity even in a parallel incident beam. On the contrary, diffraction by the sphere means that the direction of the scattered wave is changed by an angle of the order of X/R. This has the consequence that the shadow is dissolved by the diffracted beam in a distance of R 2 /X from the scattering center, and the elastic scattering can there be observed. The angular distribution of the shadow scattering is easy to compute for a black sphere, in the limit X/R « 1. The result is that da(8, <p) _ R 2 J x 2 (kR sin 8) da ~ ~ shTf? (46) • where k = 1/X = h/M v , R is the nuclear radius, and <p the scattering 58 A Survey of Nuclear Reactions [Pt. VI angles, and.Ji(z) the Bessel function of the order of unity. Actually the contributions of partial waves with I ~ R/X are small but complicated. If K/R ~ 1, the effect of important values of I will not be given correctly by (46), and the distribution and magnitude of the elastic shadow scatter- ing will be more complicated. The success of this result has suggested a nuclear model of optical type in which the nuclear volume is regarded as a sphere of "gray," not black, material which both absorbs the inci- dent wave in part and disperses it, as a result of the phase shift due to the mean potential which the incident nucleon feels in passage. This model has given interesting results in the high-energy region, which is discussed in more detail in Section 11. It is worth pointing out that the maximum value of Caba^ is (21+ l)wX 2 , and that this maximum can be reached only if y = 0. The maximum value of <r e \ w is (21 + l)47rX 2 and can be attained only when y = +1. There is in general a range of values of tr e i (i) possible for each value of (Tabs^. That the maximum scattering cross section is four times the maximum capture cross section reflects the fact that maximum scattering comes from just reversing the phase of the incoming wave to form the scattered outgoing wave. This has the effect of adding to the plane wave just twice its outgoing part, while complete absorption simply removes the outgoing portion of the plane wave. Thus the cross section for scattering depends on the square of twice the amplitude correspond- ing to complete absorption, which gives four times the cross section. 2. Influence of Extranuclear Forces. We have so far considered only maximum and minimum values for scattering and absorption, obtained by assuming values for the quantity -r\. It is clear that a real calculation of 17 would require the solution of the many-body nuclear problem, 1 and this is precisely what we are unable to do. But we can try to bring explicitly into view the effect of extranuclear forces, reserving £ to de- scribe the effect of the specific nuclear interaction and the formation of the compound state. The nucleus, in absorbing the incident beam, changes the intensity of the outgoing portion of the wave. If there is no absorption, J 77 j = 1 and we can write i\i = e 2is l and | fe | 2 = | y out | 2 . With absorption, some of this wave is removed, and we can write | ^ out | 2 = | Win | 2 — I ^ remove d | 2 . But we will regard the wave removed as altered by the nucleus proper only at the nuclear surface, writing then : Wout I 2 = I Win | 2 - f\ ^urf | 2 = I Win | 2 - f\ Win | 2 ■ ^^ (47) I Win I 1 An important paper (W7) is based on an effort to throw the whole burden of describing the reaction onto the quantity jj. We shall discuss this in Section 6. Sec. 5C] The Course of Nuclear Reactions 59 with the quantity / representing that fraction of vW which is removed by the effect of the specific nuclear forces. But we have already shown, in (35), that we could write | ^ surf j 2 /| ^ in | 2 = P. With Eqs. (44) in mind, we get I *o»t I 2 = I * ia | 2 (1 - fP) a abs ~ U in | 2 - I ^ out | 2 ~f P \ fc„ J2 (48) where we can regard f as an expression of the sticking probability £, discussed above. This would allow us to extend the notion of the stick- ing probability, as some authors have done, even to reactions where the statistical notions were not strictly applicable, in order to give a kind of transition between statistical and other views. We then can take as a fundamental relation for the cross section for compound nucleus for- mation : *,-,= (2Z+l>*.- 3 -Ptf(^)-&i (49) with the barrier penetrability and sticking probability explicitly indi- cated. C. The Disintegration of Compound States 1. Competition. The next step in the nuclear reaction, the disinte- gration of the compound state, takes place with a probability D p = r p/ 2l V (41), as we have already said. We ought to recall here the na- ture of the assumption behind this way of writing the reaction: we assume that particular features of any single compound level are not important. Many states are involved, with random phase relations, either because of the poor definition of the particle energy in the incident beam, or because the states are so widened as to overlap. Otherwise, the results we give will be valid only for averages over many states or even over many similar but not identical nuclear species. The most striking consequence of the expression (41), for D p is the phenomenon of competition. Evidently once the compound state is formed it can decay in many ways, for each of which there is a value of the appropriate T,-. The energy of the compound state will determine the value of the r/s in our approximation at least. As the energy varies, given reactants may produce different products. As each new energy threshold for particle emission is passed, the corresponding r,- rises from zero, and the compound state has then another possible decay mode. Since the total cross section is limited, the rise of one Tj must reduce the yield for the others. This is called competition. For a 60 A Survey of Nuclear Reactions [Pt. VI definite example, consider the reactions induced by neutrons incident on Br 81 . The reactions which have been observed are four (S5) : (i) Br 81 (n, 7 )Br 82 (ii) Br 81 («,p)Se 81 (hi) Br 81 (n,a)As 78 (iv) Br 81 (n,2n)Br 80 At very low incident energy (thermal neutrons) only T y is different from zero, and the first reaction takes place exclusively. As the energy in- creases, the threshold for the second reaction is reached at a couple of Mev, and it begins to appear. At still higher energies the latter two reactions are possible, and their competition is noticeable in the cross section of the (n,p) reaction. It is clear that the computation of <r„ for the formation of the compound nucleus will apply to all these reactions (though the resonance neutron capture will need special treatment) and that the values of the individual cross sections can be obtained if only the relative T's are known. 2. Specific Level Widths for Particle Emission. We have already de- scribed, in Eq. (33), how the widths are related to cross sections, and we have given formulas for the computation of individual widths. In practice, of course, we want not so much the width for emission, say, of the proton in reaction (i) above, leaving the Se 81 nucleus in a given state, but rather the total width for the entire range of proton energies and residual nuclear states which are available. Of course, we have but to build this up out of the various individual widths, and we shall obtain at the same time the often interesting energy distribution of the out- going protons. From (33) we have the value of T pa for the emission of a given particle with definite energy (averaged over many levels). If now the residual nucleus can be left in many possible states a, so close that they too can be represented by the statistical expression for the level density, we ob- tain, for the total width for particle p, T p = ^ T p i a , which we can write l, a in favorable cases as an integral: 1 /»Ej, max m fi — - — • ~-j<7 p {E)p R (E™**-E p )dE p (50a) r P = Here all factors are known: a p is obtained from suitable use of (49), the level densities are given by some model, taken say from Eqs. (31), and only the factor £ which occurs in <r p expresses the model's lack of Sec. 5C] The Course of Nuclear Reactions 61 preciseness. (We suppress the weight g R , including it in the value of PR; the level density.) We hope to find the £ variation small, and the value of £ near unity except in special cases. Thus f p can be computed for charged nuclear products or for neutrons. We shall have to reserve the gamma-ray width calculation for a later section; it is almost always small compared to heavy-particle widths as soon as the emission of heavy particles is energetically allowed. For the computation of D p clearly the factor l/ Pc (W) is contained in all terms r y , and the result is a function of E p maK only. The dimensionless integral of (50a), without the factor \/t? Pc , is referred to in the literature (W6) as the / P (-Ep max ) for a given state (S5). Under specific headings we shall discuss the calculation of these /'s in more detail: / P (^ max ) = ir 2 Pc(W)T p (50b) 3. The Thermal Analogy: Temperature, Cooling, Evaporation. From the definition of the function f p it is seen that f p is just the width for emission of particle p from the excited compound nucleus, measured in terms of the mean spacing of levels at that excitation of the compound nucleus. There is in addition the numerical factor w 2 . For this reason we shall refer to the dimensionless / function as the specific width for emission of particle p. It is interesting to view the process of emission of particles from the compound state as a kind of thermal evaporation. Just as a water mole- cule may evaporate from a drop of water, so does one of these nucleons leave the excited nucleus, in which the excitation energy plays the role of thermal agitation. Equation (50a) leads to an expression for the energy distribution of the emitted particles, which is of course just the integrand in the expression for T p . We have for I(E P ) dE p the relative number of particles p emitted with energy E p in the interval dE p : I(E P ) dE p = I E p <t pPr (E™* - E p ) dE p (51) where the constants have been lumped into 7 . Now from Eq. (30) we can write the level density PB in terms of the entropy of the nucleus, S. We get _ IpEpffp exp [SnjEjT* - E p )] T(2irdW B /dT) y * Now we shall make an approximation which is justifiable only for sufficiently high values of the excitation energy of the residual nucleus (£p max - E p ). We shall expand the entropy about the maximum value 62 A Survey of Nuclear Reactions [Pt. VI of excitation energy which can be left behind in the residual nucleus, namely about E p ma *. We obtain then dS S(W B ) = S(E™* - E) = S(E p ma *) - E p ■ dE p _ r Now, remembering that the exponential varies much more rapidly than the factor l/(dWR/dr)^, we use the familiar thermodynamic relation dS/dE p = 1/r. Inserting this, we get (IE P ) = const X Ej p<r p expl I (52) Here r is the temperature (in energy units) at which the excitation energy of the residual nucleus is on the average Wr = E p If the function ^ a fc) \ Neutrons c 1) S N. Protons x 10 <D > '+3 "/ s ^^^ £ / / ^^»^ i ^ ^^^k. ~3 / ^*r ^^ « ' ^^ i i ' ^> 4 8 12 16 E p , kinetic energy of emitted particle (Mev) Fig. 6. Energy spectra of evaporated nucleons. Note reduction of proton evaporation due to Coulomb barrier. Epff p is not too rapidly varying, as it is not for neutrons of some Mev energy, for example, the emitted particles have exactly the Maxwell distribution in energy, just as do the molecules evaporating from a drop. The temperature T corresponds to the temperature of the drop after emission of the particles — the temperature of the residual nucleus — which is not the same as that of the compound state, for of course the single emitted particle has a non-negligible fraction of the excitation energy. This makes the evaporation analogue somewhat less exact; we must think of a drop with only a hundred water molecules. If the particles p are charged, the cross section will be quite energy-sensitive; low values of E p will mean that the penetrability of the barrier is low, and will distort the Maxwell distribution, shifting the maximum to high energies (see Fig. 6). Sec. 5C] The Course of Nuclear Reactions 63 If the variation of a p with E p is neglected, the maximum energy of emitted neutrons will lie at E p = r. Table 4 gives temperature as a function of excitation energy for several nuclei; these temperatures were obtained by using the level density expression of (31). That they are clearly rather small compared to the excitation energy helps justify the approximations made. This means also that the outgoing particles TABLE 4 Nuclear Temperature in Mev, Atomic Number and Excitation Energy A- = 80 120 160 200 240 Excitation Energy (Mev) 2 0.61 0.51 0.46 0.43 0.41 5 0.96 0.81 0.73 0.68 0.65 10 1.36 1.15 1.04 0.96 0.91 20 1.93 1.62 1.47 1.36 1.29 will in general take away only a small part of the excitation energy, leaving behind an excited residual nucleus, which may in turn boil off still another particle, and so on. It is this process of gradual "cooling" by "evaporation" which is often the origin of the complicated multiple processes listed in Appendix II. 1 It is not easy to justify the two assumptions: (a) the statistical treat- ment of the level density, and (b) the neglect of E p compared to E p max . The level density formulas certainly overestimate the density of the lowest levels, or at least the lowest ones accessible from a compound state experimentally defined. Certainly the formula will not be very reliable below E p m »* of 5 Mev, and for # p max - E p of less than a couple of Mev. Above that it should be fairly reliable, until the excitation energy reaches values of 10 Mev or more per particle. The lightest nuclei will of course make difficulties for the statistical treatment. The release of neutrons from the moving fission fragments by evapora- tion is to be expected and should account for the prompt fission neutrons, and a spectacular example of "boiling off" is found in the high-energy fission of bismuth, where apparently the most probable fission fragments indicate that the compound nucleus which divides by fission is 12 mass units lighter than the state initially formed. This implies the loss of many neutrons by something very like the evaporation we have de- scribed. The nuclear "stars" of cosmic rays often represent such proc- esses. We shall discuss them at greater length in Section 11D1. 1 See, for example, (T5). 64 A Survey of Nuclear Reactions [Pt. VI SECTION 6. THE DISPERSION THEORY: RESONANCE REACTIONS The statistical theory which we have described has been based on an avoidance of reactions in which the properties of a single compound state are prominent. But some of the best-known and most interesting nuclear reactions are exactly those in which spectacular resonances indicate the importance of the properties of the individual states. Starting from the idea that such levels must be considered in detail as individuals, several authors (B13, B21, K2, S9) have tried to find a general quantum-mechanical solution of the many-body problem pre- sented. They based their treatment on the same two-step notion of the reaction as does the statistical treatment of Bohr. The compound nucleus is the system of incident particle plus target, a system whose eigenvalues are not real and whose eigenstates are not stationary but widened and quasi-stationary, as we have described them. The forces of interaction between incoming particle and target cause transitions between the initial state of the incident particle and an intermediate state which is one of the widened states of the whole system. Then this state, by virtue of the internucleon and nucleon-radiation perturbing couplings, itself decays to the state which contains outgoing particle and residual nucleus. This description was called the "dispersion theory" because of its formal near-identity to the calculation of the optical dispersion of atoms, in which the incident quanta are first absorbed by the atom, and then another quantum, perhaps the same in energy as the first, is re-emitted by the excited atom. The initial and final states of the nuclear system, then, are combined only by the mixing through the compound state, and not with any considerable probability directly. This is the consequence and of course the motivation of the idea of the compound nucleus. The "mixing" is strongest when the energy of the initial system coincides with some more or less well-defined energy level of the compound state. It is of course the intention of this form of theory to give a complete account of nuclear reactions, and the results of statistical theory are expected to follow from dispersion theory calculations when suitable averages are taken, and suitable assumptions made about the relative phases of the wave functions involved and about the character of the levels of the compound state. Whenever really sharply defined incident energies are experimentally available, and whenever the states of the compound nucleus are reasonably well- defined, the dispersion theory ought to yield more information than the statistical model, however improved. Sec. 6A] The Dispersion Theory: Resonance Reactions 65 The program of the dispersion theory is too ambitious for full success. The method of perturbation theory which works well for the weak radiation coupling of the electrons in an atom cannot be expected to give, even in theory, an adequate scheme for the calculation of the result of the strong interactions among nucleons. Wigner and co-workers have given a beautiful general theory (see Appendix II), if a rather complicated one, which does not employ the ideas of perturbation theory but insists only that the nuclear forces act within a well-defined and not too large region of space. Using the ideas of the ordinary Schrodinger wave equation, we shall indicate the physical connection between a simple one-body model of nuclear reactions and the most useful results of the complete dispersion theory, but for full discussion of Wigner's S-matrix treatment the literature should be consulted. A. The One-Body Model and Its Difficulties The principal notions of nuclear reactions before the early 1930's were based on a model much simpler than the one we have been dis- cussing. It was built up largely from a study of the decay of alpha- radioactive nuclei and the light nuclear resonance reactions with alpha- particles. The picture was that of the Hartree model of the nucleus, i.e., that the many nucleons produced a net potential well, a combined force field in which the particle to be studied moved. In alpha-decay the alpha-particle shuttled back and forth in this well, until, once in a while, by chance it leaked out of the barrier by the familiar "tunnel effect" of quantum mechanics. For scattering, the nuclear forces supplied a potential well in the same way. To account for the resonances observed, the charged particle was thought of as penetrating the external barrier and moving into the potential well; when the particle wavelength was just such that the particle could produce a standing wave in the well by virtue of in-phase multiple reflections, the particle energy was in resonance with one of its possible stationary states in the potential well. Constructive interference built up the wave function strongly, and absorption grew very rapidly at several well-defined energy levels. This worked quite well for such reactions as Mg 24 (a,p)Al 27 . It seemed indeed to be a reasonable model. The first slow-neutron resonance work showed up its major weaknesses. It became evident that in this theory scattering and absorption cross sections should increase together. Barring unusual selection rules, the increased particle amplitude inside the well should lead to increased re-emission — scattering — no less than to increased absorption. Even if capture were somehow prevented for some scattering resonances by peculiar effects, which might limit the 66 A Survey of Nuclear Reactions [Pt. VI dropping down of the particle through radiation to a lower level in the potential well, it seemed hard to understand why every level which permitted capture would not yield an even stronger resonance scattering. The simple analogue of the absorption of sound of the resonant frequency by a Helmholtz resonator is complete. The constructive interference of the wave reflected within the resonant cavity builds up its amplitude; sound energy is not only absorbed, but also strongly re-radiated. Yet experiment showed that strong absorption is not accompanied in general by strong scattering. We have already formulated the general problem of scattering and absorption by a nuclear center of force. Let us inquire into the results of a one-body model, using this formalism (Eqs. 41 et seq.). Consider the case appropriate for thermal neutrons, with wavelengths very large compared to nuclear dimensions. Then we can write for the radial part of the wave function in the region outside the nuclear radius R * = 1 u(r) (ti)^~ «( r ) = V e ikr + e- ikr (53) Only the 1 = partial wave is important, and outside the radius R there are no forces. From Fig. 7 we can see that this external wave function r^ oat = ve ikr +e- -V Fig. 7. Wave function for scattering of thermal neutrons from nuclear potential. must be joined to the internal solution. For definiteness let us consider that the internal potential acting within the radius R is represented by a rectangular well. This is in no way essential for the result, but it simplifies calculation greatly — at the expense of realism. Within the nucleus, then, the neutron has the wave function ^int = u(r) u(r) = A cos (Kr + e) (54) Sec. 6A] The Dispersion Theory: Resonance Reactions 67 with K some wave number, much larger of course than the wave number * of the slow neutron outside the nucleus, and e some phase constant The boundary condition at the origin will demand that e = nr/2, with n odd. The wave equation now requires that the constants be so ad- justed that both ^ and ty/dr remain continuous across the nuclear boundary. Let us define the quantity /, the logarithmic derivative of the wave function at the nuclear edge made dimensionless by a multi- plicative constant : / = R (du/dr)/u | r=K (55) Then the continuity conditions are both satisfied if we require / in = f outl (r,e ikR — P ~ikR\ -KR tan (KB + e ) = ikR — i One %kR + e~ ikR ) Solving for rj, we obtain -2ikR (#in - k R) Win + kR) ^ 7) Recall expressions (44). They give (56) ffrf » (0) = p d ~ I " I 2 ) and ,/) = J: (| i + , | 3) k 2 If we find v , both cross sections are determined. Clearly we have not yet allowed for the possibility of absorption— we have used only one level within the well-and our | „ | 2 = 1. This follows from the form of (57) and the fact that / in is purely real. Now we make use of the small value of kR « 1 for slow neutrons. We can write (57) as _ (-if + kR)(if - kR) fc 2z-,2 '= fTm- *" d [1 + F ^'f^ by expanding. Then the cross section becomes <r^=~[l + \v\ 2 + 2ne( v )]=^.-Jf^- k 2 k 2 f + k 2 R 2 <kir 1 ~ K 2 ' tan 2 (KR + nr/2) + k 2 /K 2 (58) This cross section shows a maximum at zero energy regardless of the nuclear size. But, for nuclear sizes and internal motions such that KR + mr/2 = 0,7r,- • •, the small cross section at low energy becomes instead an infinite peak. If we fix a definite value of slow-neutron energy and imagine that the nuclear radius, say, varies smoothly 68 A Survey of Nuclear Reactions [Pt. VI strong scattering resonances will appear for special values of the nuclear radius. Somewhat more familiar-looking resonances, appearing for definite values of incoming neutron energy, would have resulted had we considered partial waves of higher angular momentum, but the general features would not have been very different. Now, moreover, we can calculate the amplitude of the wave inside the well, taking an incident wave of unit flux. Using (54) and (56), we get 1 o / n A A 2 ^ K Thus the amplitude of the neutron wave inside will become large at resonance, and in fact we can write the cross section for scattering in terms of that amplitude, at least near resonance : o- ac (0) = 4xA 2 near resonance But, clearly, the radiative transitions to a lower state will have a matrix element proportional to the amplitude A, and the probability of such transitions, and thus the cross section for absorption, will vary like A 2 , as we should physically expect from the fact that A 2 measures the time the particle spends inside the nucleus. The ratio <x a bs/o- S c will therefore show no marked change at resonance; strong scattering and absorption resonances will be found at the same energy. This is contrary to all experience. Furthermore, it is easily seen that the resonance levels are spaced in energy much too little to correspond to adding another half-wave to the inside wave function, and that the variation of cross section with nuclear radius (i.e., with the mass number A) is far too erratic to be accounted for on such a picture. The one-body model gives much better results when the external wave function determines the broad course of events, as in alpha-particle radioactivity. Where the interior state of the nucleus is decisive, the model is inadequate. The strong physical plausibility of the compound nucleus picture, added to the difficulties of the one-body description of neutron capture, gives it its present importance. The modifications which the strong effects of shell structure will certainly require, especially at moderate excitation, have still not been completely worked out. B. The Dispersion Theory for an Isolated Resonance The difficulties which surround the derivation of the theory of nuclear reactions by perturbation methods have led to reconsideration of the basis of the whole matter. It turns out that the most important features of the theory arise from the fundamental nature of scattering and ab- sorption processes themselves, which are displayed fully only in the Sec. 6B] The Dispersion Theory: Resonance Reactions 69 rather complex nuclear domain. The general theory has been developed by Wigner and several of his co-workers. We shall discuss mainly a quite satisfactory but much simpler and less general approach developed by Weisskopf and others. In it the features of the one-body model which make scattering and absorption so pictorial are generalized to fit the physical picture of a compound nucleus. In the one-body model we have shown that, for a case when only two alternatives, elastic scattering and absorption, are possible, the logarithmic derivative function / determines the amplitude 77, and hence both cross sections. But in the one-body model the complex number / (which must be complex to allow absorption) is fixed by the nuclear potential well. Both modulus and phase are given. Weisskopf regards this same quantity/, now in general a complex number, as determined by the whole structure of the nucleus. It will be different from level to level, varying in a manner much more complicated than does the one-body /, and in particular having modulus and phase independently varying. We define the resonance energies and level widths by giving the properties of/. Thus we have a kind of phenomenological compound nucleus picture into which the properties of many levels enter, but through a single function whose determination from the actual make-up of the compound nucleus we leave perforce to the physics of the future. We shall require that (1) / is a function only of the energy (and other constants of the motion) of the compound nucleus, and not explicitly dependent on the incoming particle, and (2) / is a well-defined function, defined by (55), in spite of the uncer- tainty of the value of R, the nuclear radius, at which internal and ex- ternal waves are to be fitted. Both of these conditions can be met by the requirement that the wave function just outside the nucleus vary only slowly over a distance which corresponds to the mean spacing of the nucleons within the nuclear matter. We can regard the value of R as a parameter to be chosen to give the most reasonable average behavior of /. Some values of R, and the related internal wave number k, will not work. No value will allow us to assign the behavior of / uniquely, but the best values will yield reasonable statistical agreement with the observed properties of many levels. 1. Derivation of the One-Level Formula, (a) Without External Forces. Guided by physical considerations, we shall now try a general- ization of the one-body model. The function / is defined by the ex- pression f = R (du/dr)/u \ r=R (55) We expect zeros of / when the slope of the wave function vanishes at the nuclear edge, and infinite values if the wave function itself vanishes 70 A Survey of Nuclear Reactions [Pt. VI there. Without loss of generality we can take over the special form of / from the one-body theory: /= -kR tan z(W) (59) The tangent function no longer has as argument the simple and well- defined phase of the internal wave function in the one-body potential well, kR + «. It now has instead a function z(W), which increases monotonically as the excitation energy W = Ej, in< j + Eun increases, and takes the value nx at each of the successive resonance energies W r = E b + E T , but may vary as it will in between. The results of the theory now depend only on the assumption that the variation of z(W) with energy is as smooth and simple as possible. We shall see how all the properties of the successive resonances can be described by a suitable trend for z near each resonance. The prediction of the behavior of z in detail is given up for the present theory; it could be determined only if the actual motion of the entire compound nucleus in every eigenstate were known. But the one-body picture did not explicitly include absorption. We allow for that here in the familiar way: we introduce a damping factor to reduce the amplitude of the now only approximately stationary state. The damping arises of course out of the possibility that the system can change its state not simply by decomposing to re-emit the incident particle, but in some other way. Such a damping will as always be expected to widen and depress the resonance peaks. We write the energy as a complex quantity: W = W — zT a /2 so that the energy eigenfunction becomes Now the probability of occupation of the given state decays in time with the factor e~ r " t/h \ \[/(0) | 2 , and the mean lifetime of the state is given by r = h/r a while as usual the uncertainty principle will imply that the state is defined only up to a width r a ~ h/r. This will indeed follow from the calculated resonance shape. Evidently the concept of resonance level is useful only if the energy W has but a small imaginary part, i.e., if T a /W <K 1. Although we could write the function quite generally, we shall use the approximation of expanding / in the neighborhood of the resonance energy, leaning heavily upon the smooth behavior of z(W) and the expected smallness of T. We write then _ iT a df I f(W)=f(W)- — -= +••• 2 dW\w=w Sec. 6B] The Dispersion Theory: Resonance Reactions 71 Taking our cue from the one-body model, let us define the resonance energy, E r , by the relation f(W r ) = f(E b + E r )=0 and continue to expand / near the value E r . We get f(W) = (E - E r ) dj_ dE E r iT a df 2 dE + ■ (60) E r where we use the kinetic energy of the incoming particle, E, as measure of the excitation energy W. This linearization greatly simplifies our theory. It is certainly no serious source of error provided that we look in the near neighborhood of a resonance level. How far that neighborhood extends depends of course on the variation of the argument z(W) with energy. Now we can use the relation between the function / and the phase shift r?, just as we did in the earlier model, from Eq. (57). We take the value of / near resonance from (60). If we introduce explicitly the functions fo(E) and g(E), the real and negative imaginary parts of the function /, we obtain . f(E) = f (E) - ig(E) = (E-E r )8 + ^5 (61) where we have written 5 for the quantity (df/dE)\ Er , and r o = -2g/5. Using (57), there follows from (44) kRg (62a) Cabs (0) = - (1 - | , | 2 ) = 4,rX 2 k 2 "~ '"" "" (g + kR) 2 +f 2 We can write <r so (0) similarly. The fact that resonance is marked by /o = is now plain, and the whole expression gives just the familiar one-level Breit-Wigner formula of dispersion theory; compare (62b). Now, writing the cross sections out in full, but replacing the functions /o and g by the more physical widths from (61), and defining a width r„ by the relation r n = —2kR/8, analogously to (61), we have ^abs (0) _ = xX* ■*■ n*- a m (62b) + (E - E r ) 2 (0) _ = 4ttX 2 = 4tX 2 kR i(kR + g) - f 2 1 n + e lkK sin kR E -E r + -(T n + r„) + e lkK sin k R 72 A Survey of Nuclear Reactions [Pt. VI From the results of (62) we can go much further in the interpretation of the argument function z(E) of (59). Plainly it goes through multiples of ir, giving zeros for the logarithmic derivative function / at each resonance. It is also clear that the contribution to absorption, say, of any level will fall off on each side of the resonance energy, with a charac- teristic width given by T n + T a = T. But this width is inversely (re+3)x (w + 2) 7T (n-hl)T r large small Excitation energy W Fig. 8. The argument function z(W) as a function of the excitation energy "FT in a region containing several resonances, W a , Wb, • • •, with differing widths (F7). proportional to the slope of the / function near the resonances. From (57) it is easy to show that . |2 _ 1 + a 2 Im (/) 1 " ' 1 - a 2 Im (/) where a 2 is positive definite; and, since we require from the absorption cross-section formula (44) that | 17 | 2 < 1, it follows that Im (/) < 0, and in our expansion therefore 5 must be non-positive. This justifies our use of the minus sign in the definition of T n and r o and confirms the choice of the tangent function in (59). We can now see graphically the meaning of Fig. 8, in which the function z(E) is plotted schematically in a region of several resonances. The width is given in our approxima- tion by T = — const/5 = — const -dz/dE. Since 5 is necessarily non- positive, z does increase monotonically, as we expected. Where z changes rapidly with E near a resonance, 8 is small and the widths large. We have drawn narrow levels at resonance energies a, b, c, wider ones at d and e, and a very wide one at /. We cannot predict the level positions Sec. 6B] The Dispersion Theory: Resonance Reactions 73 or the slope near those values. But we have thrown the whole burden of the determination of reaction and scattering cross sections into the behavior of a single function z, which is a kind of equivalent internal phase. If z varies smoothly, we can expect that dz/dE will be small, and the widths small when the levels are closely spaced; and reverse will also hold. By such simplifying assumptions on the smoothness and statistical regularity we can produce various results concerning the average behavior of nuclear resonance lines. Individual levels must be examined empirically, but can be fully described by this theory. The choice of the matohing radius R can be fixed by the pragmatic test of how simple and statistically unbiased is the behavior of the many levels, for R must divide the region outside the nucleus from the one within, and this division will be physically most clear cut for a particular R. Improper choice of R will allow the behavior of z to be affected by the external region and will distort the expected intelligibility of the proper- ties of z. (6) With Barrier. So far we have considered only the case appropriate to slow-neutron reactions, where the orbital angular momentum I = 0, and neither centrifugal nor Coulomb forces are present. Outside the nuclear radius R the potential is strictly zero. In this case, and in this case only, the external wave function is given not just asymptotically but everywhere outside the nucleus by the partial plane waves of (53). It is not very hard to extend the calculation to the more general case. We write the radial part of the wave function in the external potential as before, for a definite value I of the orbital angular momentum. But, as we approach the nucleus, the wave function is no longer a simple plane wave. We introduce the independent solutions of the wave equation with the given external potentials which go over asymptotically into incoming and outgoing partial plane waves. These solutions we call u t and u for incoming and outgoing parts, respectively. (The Coulomb potential can be thought of as screened very far away to avoid the logarithmic term in the asymptotic phase shifts.) We write, for unit flux, 1 Ui(r) + V U„(r) 1 e -i^r-l,l2) + + i {k r-hl2) Ur) =W* r — ^ " (63) Now, as we come in to the matching radius R, the behavior of the solu- tions Ui and u is completely known for any given external potential. Since they are adjusted asymptotically to the same amplitude, and one solution could be obtained from the other by simply reversing the direc- tion of time, one is the complex conjugate of the other, u t = u *. We 74 A Survey of Nuclear Reactions [Pt. VI shall also need the penetrability of the external barrier, which we can define, as in (35), by the relations pd) = I "^ I = i_ = i (64) M,(°0) J 2 _ 1 1_ «,(#) | 2 ~ I Ui(R) j 2 ~ Ui(R)u {R) Here we affix the I value corresponding to the orbital angular momentum partial wave involved. We shall also make use of the Wronskian rela- tion, obtained in the usual manner by writing the wave equation for each solution, cross-multiplying, and subtracting, to get d / du du{\ — I Ui u — ) = uiuj — u Ui = const dr \ dr dr / Since this is true for all values of r, we evaluate f or r — * » to obtain UiU ' - u Ui' = 2ik — = 2ikP (l) (65) U Ui Now let us define the quantities Fi i0 = RuiJ /ui i0 \ r= u by analogy to the function /. With all this not very complicated machinery, we have the solution at hand. Again we write down the matching condition at the nuclear edge, just as in (56) : /in = kR(u/ + rfu ')/(ui + r]U )\ r=B Solving for the amplitude v, we get Ui{R) Fi - f in Ui {R) (if ia - iA + y) V = = (66) U (R) F — /i n U {R) (if in - iA - 7) where we have written the expression in the form closest to (57) and have introduced the notation F,-, = A ± iy for the complex numbers Fi i0 . Writing /i n = f„ — ig as before, and evaluating y from relation (65), Fi - F = 2iy = -2ikRP m y = -kR-P w (67) we get from the fundamental relation (44) between o-„bs and 17, the result : ±irX 2 -kRP<> l) g This is very like the result for the s wave obtained in (62a), and, indeed, for I = and no external forces, «,-,„ = e ±tkB , P (0) = 1, and A = 0, which gives exactly (62a). If we write for the unimodular number Ui(R)/u (R) = e~ 2ia , which was just e~ 2%kR for the no-force case, we can write the scattering cross Sec. 6B] The Dispersion Theory: Resonance Reactions 75 section as well, and, if we introduce the same linear approximation as the expansion of (61), we have <r abs (0 = xX 2 • (21 + 1) _£=!• f68b) (E ~ E r )* + (r„ + r a ) 2 /2 {bm crj l) = 4ttX 2 -(21+ 1) in which we have set + sin ae ta T n = -2kRP«)/~ dE (E - B r ) + (i/2)(r„ + r u ) r« = -2g/,_ «,(•> (68c) and defined the resonance energy E r by the relation (E r - E r «»)8 - A = (68d) E r = E r m + - with/ (#/°>) =0 5 = — S dE _ r Equations (68) are the principal results of the one-level theory, and they exhibit a number of interesting properties which we shall discuss briefly. 2. Features of the One-Level Theory, (a) Level Shift. It will be ob- served that the resonance energy is increased by an amount A r = A/5 from the value of the energy E r m at which the value of z was set equal to nir. Since this value of the energy is in any case not observable directly, nor calculable in the present theory, the shift might be regarded as meaningless. It has been pointed out, however, that in the com- parison of the successive levels of mirror nuclei, whose energy levels might be expected to differ by an easily calculated Coulomb energy, and in no other way, the effect of A r could be seen, for the excited states of two mirror nuclei will in general be capable of different modes of decay, since thresholds for charged particle emission will differ. Thus A r will differ for the two nuclei, and the observed resonances will not show energy displacement exactly equal to the Coulomb energy differ- ences. Qualitative agreement with the observations has indeed been found in at least one case (El), that of the pair N 13 , C 13 . Physically the level shift can be ascribed to the fact that in a compound level the particle which ultimately leaves the nucleus spends considerable time outside the nuclear radius and within the external force field before its departure; its wave function extends with sizable amplitude beyond the distance R. A more formal but somehow familiar description is to observe that such a shift represents the reactive part of that coupled impedance whose resistive part gives the familiar level broadening and damping. 76 A Survey of Nuclear Reactions [Pt. VI (b) Reduced Widths. The particle width r„ is given by (68c) as just '-"""'(-ID" It is easy to compare this with the result of the compound nucleus ap- proach in Eq. (38). We see that here too we can define a "width without barrier," or reduced width, by writing F n = 2kRP ( - l) y r . The magnitude Y r is just the reduced width, which is dependent only upon the internal state of the nucleus, as described by the function /. Comparison with (38) shows that the reduced width is a measure of the probability of the particle being at the nuclear surface. The absorption width also can be written in this form, introducing a wave number k to represent some characteristic wave number for internal nuclear motion. We write the hitherto unspecified imaginary part of / near resonance in the form +g = icRha, and then we obtain r also in terms of the reduced width and of a dimensionless expression giving the imaginary part of /: r a = 2KRh a y r The observed width of a level is not always given by these formulas, since the variation of A in (68a) as the incident particle energy is varied is not always negligible; this effect can be computed by expanding A r itself about the resonance energy in (68a), and retaining only the linear variation. For s-wave neutrons, of course, A = 0, and there is no such effect; but cases have been exhibited in which the observed width differed from the value of r = V„ + r o by as great a factor as 2 or 3. (c) Negative Peaks. Formula (68b) for the scattering cross section exhibits some very odd properties, which arise from the possibility of interference between the resonance term and the second term, called the potential scattering term. This scattering term arises from the effect of the well-defined volume in which the nuclear forces act, and also from the summed influence of all the other levels of the compound system. There is no rigorous distinction between these two ideas; in our more formal picture we can think of the potential scattering as com- ing from the fact that between resonances the value of / is such that the surface wave must be of very small amplitude, much as though the nuclear surface were the surface of an impenetrable sphere. Taking the presence of a potential scattering term as our model gives it, then, let us examine its possible consequences, at least reasonably near a single resonance level. It is convenient to observe that the resonance term, with its varying denominator which gives rise to the familiar witch-shaped peaks, can Sec. 6B] The Dispersion Theory: Resonance Reactions 77 be written in terms of a phase angle 0. With the notation %Tn r (E - E T ) + (i/2)(T n + r„) ~ A + i 2(E - E r ) we write (r„ + r„) (r» + r„) r r = r sin Be A + i cotan + i which yields the very symmetrical form: crj l) = ttX 2 - (21 + 1)| r sin QeT m + sin ae ia | 2 If we consider the resonance term alone, the familiar peak is produced by the variation with energy of the phase angle 0, going from a value of zero far below the resonance energy, taking a value of w/2 at exact resonance, and going to x far above the resonance energy. When the interfering term is considered as well, we may write the cross section in the following way to exhibit the interference : <7sc W * = — zJoi 7 = sm2 a + r2 sm2 6 + 2r sin 6 sin a cos (6 + a) Here it is clear that, if r = 0, we have pure potential scattering, which is constant with energy (for not too great changes in energy, we stay in the neighborhood of one single level) ; if a = 0, we have the pure resonant peak. The interference is described by the cross term, linear in r. A somewhat more understandable form of the expression can be obtained by a not very obvious transformation, using the trigonometric ex- pressions for sums and differences freely, and employing the iden- tity sin x + a sin (x + y) = (1 + a 2 + 2a cos y) M sin (x + e), tan e = a sin y/{\ + a cos y), which is best derived directly from the indicated geometrical construction. The transformed expression becomes r 2 r o- = h (1 — r) sin 2 a [r 2 + 4(1 — r) sin 2 a] 1A where + r[r 2 + 4(1 - r) sin 2 a] y * sin 2 (0 + tf>) 2 sin a cos a tan 24> = ^— (69) r — 2 sin a Here it is explicitly seen that the entire energy variation is contained in a sin 2 term, which recalls the general results of the method of partial 78 A Survey of Nuclear Reactions [Pt. VI waves. The phase 6 of the resonant contribution always increases by ir as the energy rises through the resonance. But the value of the potential scattering phase angle <f> will determine the shape of the observed "peak," which will be a normal-appearing peak only if <f> = 0, x, and a negative peak when <j> = x/2, with a dip-and-peak combination for intermediate values. Such "negative resonances" have been observed, and the dip preceding a peak has been found in a number of cases. The scattering maximum will be displaced from the energy value E T of the absorption maximum because of the potential scattering interference. Such dis- placements are typically rather small, and no clear example can be cited. Use of (69) and related generalized forms to examine the proper- ties of particular levels, especially statistical weights and hence spins, is frequent. 3. Spins and Statistical Weights. Up to this point we have ignored the existence of intrinsic spin for the fundamental particles involved in reaction, and of total angular momenta for the complex systems. We fix our attention on a single level of the compound nucleus. We shall assume that such a level has no degeneracies — all accidental ones being removed by coupling forces of some finite size, 'even if very small, within the nucleus — except the necessary degeneracy in spatial orienta- tion of the total angular momentum vector J. This implies, in the absence of external forces, a (2/ + l)-fold degeneracy of the compound state. But the compound nucleus can be formed in many ways. If the incoming particle has intrinsic angular momentum s, if we consider only the single orbital angular momentum partial wave I, and if the target nucleus has initial total angular momentum 7, then the total number of different ways to form a compound nucleus is (2s + 1)(2£ + 1)(2Z + 1). Of these only 2/ + 1 will correspond to the given compound level of angular momentum in question. Thus, for unpolarized beams incident on unpolarized target nuclei, and with no measurement of the spin of the resultant particle, we must multiply the cross-section formulas (68) by the statistical weight factor: 2J+ 1 ° J ~ (2*+l)(2*+l)(2J+l) (70) The absorption cross section for the familiar one-level case with 1=0 becomes, for example, 2/.+ 1 . r„r a ""abs — ~~~ 77ZZ T 7 ™ (2« + 1)(2I +1) (E- E r f + [(r„ + T a )/2] 2 In general, of course, J is not known, and several possibilities exist. Even for the specially simple case of thermal neutrons, where only Sec. 6C] The Dispersion Theory: Resonance Reactions 79 I = can contribute, and s = )4, we get the alternatives I + s = J + , I — 8 = J _, and the g J± factor is ambivalently J[l ± 1/(27 + 1)]. Sometimes J values can be assigned by study of the cross-section magnitudes. C. The Generalized Theory of Dispersion : Many Levels and Many Decay Modes We have treated the theory of reactions only in the simplest case. We have considered only two alternatives: the incoming wave is co- herently scattered, or it is absorbed. In general the theory must take account of many possible consequences of the formation of the com- pound state. It may be that the energy is not near a single resonance, but lies between two resonant values; or the widths may be comparable to the spacing, so that the effect of two resonances may overlap; or simply that compound states of different J can contribute to the emission of a single outgoing wave of fixed I, if spins are present. In all these cases, as in the case where several product particles are energetically allowed, we have to take into account the various possible courses for the reaction. In a series of papers (see Appendix II), Wigner and co-workers have presented a beautiful generalization of the process here applied in the one-level, two-alternative case. The phase shift rj which described the reaction by the relation <rj 0) = (x/A; 2 )(| 1 + v | 2 ) is generalized by introducing a unitary matrix U, such that ^f = ~-\(U~l) if \ 2 where i, f index initial and final particles, not only as to type, but also as to internal state (excitation), spin orientation, and relative orbital angular momentum. The theory now produces values for the matrix elements of (U - 1), often called the collision or S matrix, between all the states representing the various alternatives. Unlike the f(E) of our phenomenological theory, the matrices are given explicitly in terms of the Hamiltonian describing the interaction of all the nucleons in the total system. Progress is made, however, only by the demonstration that much of the behavior of the cross-section formulas can be studied by knowing precisely only the interactions outside the nuclear radius, and then replacing a detailed knowledge of the interior by certain bound- ary conditions on the nuclear surface. This is in strict parallel with the progress of the derivation we have given. Indeed, the more general method differs mainly by the complete generality into which it has been 80 A Survey of Nuclear Reactions [Pt. VI cast, and by the somewhat more complete dynamical specification of the quantities which give the important results. The theory places the assumptions of our point of view in full sight, and demonstrates that the chief features of a theory of reactions come from the ability to specify with more or less definiteness a surface which can divide a region in which the Hamiltonian is fully known from one where it is not. We shall not further discuss the more general theory here, but refer instead to the literature. Here it is appropriate, however, to indicate one entirely formal scheme of generalizing our formulas, like (68), to include explicitly the properties of more than one level. It will be seen that there is full equivalence between representing the behavior of the cross sections between reso- nances (i.e., where more than one state must be taken into account) (1) as we have done it, following Weisskopf, by the use of a function f(E) whose properties, however, cannot be simply given except in the neigh- borhood of one level; or (2) by considering the summed contribution of a large, or strictly an infinite, number of levels, whose phase relationships and individual widths and locations can be known only in principle. In the absence of a detailed solution of the eigenvalue problem of the whole compound nucleus, no dispersion theory approach which is not statistical in nature can give useful results except in the neighborhood of a single level, or at most of a small number near-by, whose properties can be approximated. The formal equivalence of the two points of view comes out clearly by examining (66). We can write the denominator factor in this way: 9(E) = r /T1N * - = £ fUE) - (A - n) j Sj - E This is an identity where the sum is to be carried out over all the poles «,- of the function g, i.e., the zeros of its denominator. Placing the sum into (62a) and remembering the linear expansion approximation, we can write the simpler formula of (62b) as Cabs — tX ZdY'iY)^/ E - Ej + \ (iv + IV) (71) The virtue of this way of writing the formula is that it exhibits the possible interference of the contributions of many levels. It is this interference which must be taken into account to explain angular dis- tribution of reaction products, especially in light nuclei, where broad levels are the rule. Here phase relations are evidently decisive. But the existence of constants of the motion, such as angular momentum Sec. 6D] The Dispersion Theory: Resonance Reactions 81 and parity, will imply selection rules. The mode of formation of the compound nucleus can influence the phase of the matrix elements which occur in the (IV) H of the many-level formulas. In those cases the basic simplifying assumption of the independent decay of the compound nucleus will not be valid. When, as above, the total cross section, integrated over all angles of product emission, is calculated, any interference terms arising between compound levels which have different values of the constants of motion, total angular momentum, or parity drop out. However, for differential cross sections experimentally given by angular distribution or correlation measurements, these interferences are decisive. Indeed, every case in which the angular distribution of some product exhibits asymmetry with respect to a plane normal to the beam axis must arise out of such an interference between compound levels of differing parity. The best-studied example is that of the gamma-rays from proton capture in Li 7 near a strong 440-kev resonance. No general form of the com- plicated formulas involved is available, 1 though special applications occur quite completely worked out in the literature. The T r 's are best represented as matrix elements between the states involved, and close attention must be paid to the coherent I states in the incident beam, and to the various combinations of I, s, s z which can give rise to each com- pound level of fixed J r and parity. D. Statistical Estimates It is interesting to look at the value of o- abs (0) in the case contemplated by the statistical theory. We think of many levels contributing in a region of energy AE at E, with AE/E « 1 ; and we examine the average absorption cross section CT abs giving only average values for the properties of the individual levels. Formula (62) gives the contribution of each level. Replacing the quantity (E - E r ) by a variable of integration, and using the fact that the width V is small compared to the region AE so that we can treat X as constant over the level, we get, for each level, <W 0) &E = 27r 2 x 2 r„r a (r n + r a ) and, for the entire set of N levels in the interval AE, . £ 2x 2 X 2 IYiy 2T 2 K 2 NT n T a (Tabs AtL = 2_, *■ n ~r *- a I n T la 1 For an interesting account of the general properties of angular distributions see (Y3). 82 A Survey of Nuclear Reactions [Pt. VI We then write for the number of levels its mean value, N = AE/D, where D is the mean level spacing, and we obtain ... 2jt X r n T a _ <W 0) = — — — r = r a + r„ (72) It is clear that this expression is the same as the appropriate form of Eq. (40), with (49): (statistical) ^o ^ ^ .. .. o ^a „ 25rT n ffab. *•"■""*" = xX 8 •!).,». Pq« = tX 2 r d where the sticking probability is now 2irY n /D, which is proportional to k ~ v at low energies. We have shown how the resonance contributions of the one-level dispersion theory sum to the statistical form if D >>> r, as we expect. Before we leave the dispersion theory, we shall cite one useful result which follows more directly from the generalized theory, or from the perturbation theory than from our phenomenological approach. The (reduced width) **, y r Vi , is shown in the generalized theory of Wigner (Jl) to be expressible as an integral over the nuclear surface of the product of two wave functions, one corresponding to the interior state of the nucleus, the other referring to the external motion of the product particle. The reduced width depends explicitly not only on the state of excitation of the compound nucleus, indexed by r, but also on the nature of the state of the residual nucleus, and therefore on the quantum state of the emitted particle. We have neglected the latter dependence, saying that only the external motion of the emitted particle was important. If, however, we sum over all the possible states / of the residual nu- cleus, we obtain a limit for y r which cannot be exceeded. This limit may be approximately evaluated by using the familiar sum rule for the product of the two matrices, y r / y2 , and regarding the internal wave function as expressing the fact that the nucleus has a roughly constant density. The upper limit thus approximated for y r is r 3 h 2 3 Tr = i^P> - ~2^~I^ MeV ' f ° r nUCle ° n Width (73) This upper limit sometimes permits the exclusion of certain partial waves, when the low penetrabilities associated with high I would imply a y r violating the sum rule limit. The similarity between this limit and Eq. (39) is evident; from the general scheme it follows that we would expect widths near the limit for a one-body model, which would diminish as more and more levels contribute, or more and more particles take Sec. 7A] Some Typical Nuclear Reactions 83 part in the nuclear motion. In general, our phenomenological theory would lead to the result that for a smooth enough variation of the func- tion / the value y r /D would be roughly constant (D is the level spacing). The departure from this result will measure the significance of devia- tions from the statistical theory. SECTION 7. SOME TYPICAL NUCLEAR REACTIONS In this part we shall apply the theory of the earlier sections to some typical nuclear reaction types which it has helped to explain. In each case a much more detailed account can be obtained in the original literature; it is the purpose of this section to indicate the method of using the theory in sufficient detail, and to point out as well its difficulties and pitfalls, so that the reader may extend the few examples given, necessarily briefly, here to the whole range of experimental material. A. Resonance : The Region of Dispersion Theory Clearly marked nuclear levels, and incident particle beams with well- defined energy, are known in only two types of reactions. The most important of these (discussed also in Part VII) is the interaction of slow neutrons with nuclei, leading usually to capture, but often to scattering or even particle emission. 1 The second type is the class of reactions using charged particle beams up to a few Mev of energy on target nuclei from the very lightest up to the region of, say, aluminum. In these reactions well-marked levels can often be found (see Figs. 10 and 11), the wavelengths involved are not small compared to nuclear dimensions, and the whole approach must be based on using the maximum informa- tion about individual levels. 1. Thermal Neutron Reactions. This large class of reactions is dis- cussed in Part VII, Section 2B2. We shall here discuss it in sufficient detail to illustrate the use of the theory presented in Part VII. In the thermal region, the neutron wavelength is very large compared to nuclear dimensions. The neutron de Broglie wavelength X = h/mv is just X = 0.045 A/(E) V2 , where E is the kinetic energy in electron volts. For energies from a few millivolts up to, say, a thousand volts, which broadly defines the region of interest in these experiments, X ranges from 2 A to 10 -3 A. Throughout this range, the partial wave corresponding to I = 0, the so-called S wave, alone will be effective in reaching the nuclear surface. The penetrability of all other partial waves is so greatly reduced by the centrifugal barrier that they can be 1 See, for example, the collection of results and references in (G6). 84 A Survey of Nuclear Reactions [Pt. VI neglected. From this follows the isotropic distribution of scattered thermal neutrons and of capture gamma-rays in the center-of-mass system. (a) Effects of Target Motion and Binding. It is well to introduce here a note of caution. The usual neglect of the chemical forces acting on the target nucleus, of the thermal or zero^point motion of the target nucleus, and of the possible coherent scattering from the neighboring nuclei of the target material cannot be justified in the thermal neutron reactions. The coherent scattering of neighboring nuclei (discussed in Section 5 of Part VII) will show up strongly in cross-section measure- ments as long as the neutron wavelength is near the values which fulfill the Bragg condition rik — d sin 6 for the lattice spacing d of the crystal or microcrystals involved. This means that, in the range above 0.1 ev, the effect is not important for most target materials. The molecular binding effect is very important in determining the energy loss upon elastic collision, but again does not affect cross sections much in the region where the energy of the neutron is greater than the smallest vibrational level difference in the target molecule or crystal; and in addition this effect is of small importance for reasonably heavy nuclei. We shall treat here briefly the effect of the thermal motion of the target nucleus — the so-called Doppler effect. Let the velocity of the neutron be v in the laboratory system, and the component of the nuclear velocity toward the neutron beam be V. Then the relative velocity of neutron and nucleus is (v + V), and the relative kinetic energy fYh #rei = - (v + V) 2 = E n + (2mE n ) y *V (74) Jt to the first order in V/v. Here m is the neutron mass, E n the neutron kinetic energy. If the target atoms were those of a gas, they would move with the Maxwell distribution, giving for the fraction / of atoms having the velocity component V, with atomic mass A, "■ v > dv '(^rT e ~ m '" ir (75 > Inserting expression (74), we find, for the probability of a given E ie \, f{E iei ) dE tel = — • e-t*--™ "> — D = 2 [~~j-) (76) where D is the "Doppler width." The cross section is given by the one- Sec. 7A] Some Typical Nuclear Reactions 85 level formula written in the form appropriate for neutron capture (or scattering), near the resonance energy, E r : Un <r(E„) = (77) 1 + [(E n - E r )/(T/2)f K " } and E n here is of course to be replaced by the relative energy i? rel . The term {2mE n ) V2 V is the correction for the motion of the center of mass. With this we obtain, for the effective cross section S(E n ), 5{E n ) = f*(E Tel )f(E Tel ) dE rel = c F (-' x) with x = (E n — E r )/(T/2), the deviation from resonance in units of the total width, and the integral \D / 2(tt)^Z>J_ e -rHx- X ') 2 /iD 2 ^ x '_ 1+x' 2 This function simplifies for large natural width to the form of (77) exactly as without temperature motion. But, for small natural width compared to Doppler width, the cross section at resonance is changed and the shape of the curve altered. In this case, with T/D <K 1, we obtain simple expressions for both : (1) E very near the resonance energy, (E n — E r )/D « D/T, when (*•)* r _ . 6 -(je„-fi.)7.D s 2 D (2) E very far from resonance, (E n - E r )/D » D/T, and 1 ^ 1 + [(E n - E r )/(T/2)] 2 the value without Doppler motion. For E n = E r , exact resonance, we can easily obtain \D / 2 D \ (tt) 1a J ) If T/D 3> 1, this reduces to F = 1, and the cross section at resonance for large natural width is just a(E r ) = <r . For exact resonance, and small natural width T/D « 1, F(T/D, 0) = (r) K /2 • T/D, and the measured cross section at resonance becomes (»•)* r 86 A Survey of Nuclear Reactions [Pt. VI much reduced by the Doppler broadenings. Lamb (LI) has shown that these same formulas hold even if the target — as is usually the case — is not a perfect gas but a Debye solid (other models will give similar results). In fact, if we replace D in (76) byanewvalueD = 2(£ , „-/cre q ) M , where T e(l is the equivalent temperature corresponding to the mean energy per vibrational degree of freedom (i.e., kT e<1 = mean vib. en- ergy/vib. degree of freedom), then the formulas are unchanged. This holds as long as either the natural or the Doppler width is large com- pared to the Debye temperature. For a not too sophisticated application of the Debye theory, the equivalent temperature is simply related to the Debye temperature. A few values are shown in Table 5. For other TABLE 5 Equivalent Temperatubes or Cbystal Vibrations T eQ /T 2 /"Debye 1 2 1.06 1 1.15 0.75 1.35 0.5 1.8 0.25 cases, consult Lamb and the other references. It will be observed that all this discussion applies only in the case where the wavelength is such that neither crystalline nor molecular diffraction effects are appreciable. (6) Level Widths and Positions. With such conditions in mind, we can look into the Breit-Wigner one-level formula for neutron resonance absorption or scattering. From (62) or (70) we obtain a-X 2 / 1 \ a(E) 1A \T a forabs °abs,so = -— - I 1 ± — — — 7 ) — — TS : TTS X 2 \ ± 2/ + 1/ {E - E r ) 2 + (r/2) 3 " [a(E) « for sc (78) Here we have written for the neutron width, r„, the value r„ = a(E) 1A , and we assume that the radiation width r a is constant. These two simplifications follow from the most marked feature of slow-neutron work: the fact that we are here studying, with resolution of a fraction of an electron volt, a very small portion of the level spectrum of the compound nucleus, some 5 to 8 Mev excited. We can safely regard all factors in the widths which depend on the compound nucleus as constant over the whole resonance, and from the formula we have the results employed. A few thermal neutron reactions are exothermic for heavy particle emission, as for example those with neutrons on H 3 , Li 6 , B 10 ; N 14 . Here Sec. 7A] Some Typical Nuclear Reactions 87 the widths are very large on the scale of thermal energies, many kilovolts at least, and the cross sections become simply <r a b s ~ X 2 (E) y * ~ 1/(E) lyi . This is the famous 1/v law. Even in heavier nuclei where gamma-emis- sion dominates, the 1/v behavior appears for a range in neutron energy small compared to the energy of the lowest-lying resonance. Only where the resonance lies very near zero energy (either above it as in Gd 157 or below it as in, say, Hg) does the 1/v law fail at low energies (apart from crystal and chemical binding effects, of course). A great body of experimental information has been obtained (and compiled in useful form by Adair (Al)) on the shape of the transmission curve and from it the cross sections for very many isotopes. These yield on analysis (for the narrow isolated levels for middle to high A) fairly reliable values for the three most characteristic parameters: the resonance energy E r ; the total width T and the neutron width evaluated at 1 ev energy; a, taken from the measured cross section at exact reso- nance, c(E r ). We tabulate a few typical values in Table 6. TABLE 6 Some Slow-Neutron Resonances and Their Properties Nu- E r r aX 10 3 CO Remarks Refer- cleus (ev) (ev) (ev)* (barns) ences Na 23 3,000 ~170 r« < r„/ioo 550 sc 1 Mn 55 300 ~10 ~4,000 sc 2 Co 69 115 2-5 ~12,000 sc 2, 3 Rh 103 1.21 ±0.02 0.21 0.5 2,700 abs 4 Cd 113 0.176 ±0.001 0.115 2.2 58,000 abs 5 In 116 1.44 ±0.03 0.085 2.2 27,600 abs 6 Sm 149 0.096 0.074 2.0 110,000 abs 7 Eu 163 0.54 0.15 1.7 20,000 abs 8 Gd 167 0.028 0.12 4.5 290.000 abs 9 Au 197 4.8 <1 >3,000 abs (unre- solved) 10 1. C. T. Hibdon el al., Phys. Rev., 77, 730 (L), (1950). 2. F. Seidl, Phys. Rev., 75, 1508 (1949). 3. C. Hibdon and C. Muehlhause, Phys. Rev., 76, 100 (1949). 4. R. Meijer, Phys. Rev., 75, 773 (1949). 5. L. Rainwater el al,, Phys. Rev., 71, 65 (1947). 6. B. McDaniel, Phys. Rev., 70, 832 (1946). 7. W. J. Sturm, Phys. Rev., 71, 757 (1947). 8. L. B. Borst et al., Phys. Rev., 70, 557 (1946). 9. T. Brill and H. Lichtenberger, Phys. Rev., 72, 585 (1947). 10. W. Havens el al, Phys. Rev., 75, 165 (1947). 88 A Survey of Nuclear Reactions [Pt. VI It is seen that for all good absorbers the width r is indeed much greater than the neutron width, a(E) 1A . It is moreover reasonably constant among nuclei. This reflects the fact that it is made up of a sum of partial transition widths for the emission of gamma to all the possible lower states of the compound nucleus, which tends to average out fluctuations. But recent measurements on capture gamma-rays do appear to demonstrate a greater individuality among capture gamma- ray spectra than this point of view would lead one to expect; if those results are correct, we are seeing again a reflection of the special features of nuclear level spectra, perhaps arising from the importance of shell structure. A few examples of resonance scattering have been observed in some detail, both for slow neutrons, with I = 0, and for faster ones. The interference between potential and resonance scattering is plainly seen, along with the expected dip in the cross-section curves. Most thermal scattering, however, is simply scattering far from resonance — that contributed by the potential scattering term. The cross sections ob- served fluctuate more or less widely about the value 4x.ffi 2 , where R is the nuclear radius. The nuclear boundary cannot of course be very sharply defined for this process. Near-by levels will cause deviations which arise from the interference of other resonance terms. For the lighter elements, below A about 100, the level spacing is large, and the possible neutron widths therefore also larger (see Fig. 8) ; we may expect some sizable resonance scattering effects to show up, even rather far from resonance and without sign of much capture. The actual resonance scattering observed in Co and Mn is then complemented by some rather large deviations from the expected value, without evidence of actual resonance, in Cu, Ni, Fe, and a few other nuclei. Mn itself shows an abnormally low value of tr sc at thermal energy (H9, S6). 1 The magnitude of the scattering cross section cr sc is measured of course by transmission. Recently experiments which make ingenious use of molecular and crystal coherent scattering effects have been applied to observe the phase change on scattering. These results demonstrated that most heavy nuclei showed a change in phase of w between incident and scattered wave (F4). This would be expected from the model we have used, for, wherever the "potential scattering" term was the most important, the phase change ought to be just that produced by an impenetrable sphere. The extension of the ideas of the one-body model made by the method of Weisskopf et al. still leaves unchanged many of the simple conclusions from that picture! 1 This whole treatment follows closely that of (F7). Sec. 7A] Some Typical Nuclear Reactions 89 The high distribution of high capture cross sections through the periodic table reflects the notion that the presence of a resonance level just at the excitation energy of the compound nucleus is a matter of sheer chance, since we have magnified a single half-volt region out of many millions. This consideration allows us to ascribe the fluctuations — which are indeed wild — to simple chance positioning of levels, and of the general trend to the gradual increase of level density. The cross sections for capture slowly rise, with fluctuations of course, until in the rare earths and beyond high cross sections are common. The sharp decline in the cross section of lead and bismuth is attributed to a decrease in neutron binding energy, also implied by the general kink in the mass- defect curve which is responsible for the natural radioactive elements. The level density, which ought to increase as the particle number in- creases for a given excitation energy, actually declines, partly because of the smaller excitation energy yielded by neutron capture, and partly because of the influence of closed shells. This is another reflection of the presence of structural detail superimposed on the statistical behavior of the nuclear drop; compare Fig. 17. 2. Charged Particle Reactions on Light Nuclei, at Moderate Energy. The special ease of defining zero kinetic energy, which makes slow neu- trons so nearly monoenergetic, cannot apply to charged particle reactions. Slow protons are not very hard to make — hot atomic hydrogen — but they obviously will penetrate the Coulomb barrier so little as to make them useless for study of nuclear reactions until the temperature reaches that of stellar interiors, ~10 7 °K, where indeed thermal protons are of the utmost importance in inducing nuclear reactions, and are the agents of the release of stellar energy. But beams of charged particles up to a few Mev energy can be produced which are well-defined in energy and direction. Indeed the latest techniques * allow an over-all resolution, including the effects of the slowing of the protons in the material of the target, of the order of 100 ev out of a couple of Mev. For nuclei and excitation energies in a range where charged particles of this energy will penetrate the barrier, and where the level spacings are large compared to such a figure, we may expect to apply the dis- persion formalism to the computation of excitation functions. It will be clear that the general trend of all such reactions, which are primarily those with protons and alphas (or any charged projectile) on nuclei up to A ~ 50, will be governed by the effects of the Coulomb barrier. This will determine the width for entry T p and for re-emission of any charged particle. But, superimposed upon this easily understood 1 See the work of the Wisconsin group as described in (H7), for example. 90 A Survey of Nuclear Reactions [Pt. VI extranuclear factor will be the effects of strong resonances, which give the complex results seen in Figs. 10 and 11. We want to discuss here the method of treating such reactions; it is already obvious that only detailed attention to the properties of individual levels will be adequate, (a) The Reaction Li 7 + p. Figures 9 and 10 show the simplified experimental results for the two sets of products 2a and (7 + p')' — inelastic scattering — from the same reactants Li 7 + p. 1 The smooth rise in Fig. 9 is indeed just a penetrability curve, following the formula with a width without barrier, G, large compared to the energy variation for the protons, and an alpha-particle width which is independent of energy, since the reaction is exothermic by some 17 Mev and the alphas are far above the barrier. But the rise is not that appropriate for a pure Coulomb barrier, s-wave particles alone coming in without angular momentum; it is necessary to assume that the incoming particles pass as well through a centrifugal barrier, with 1=1. The »S-wave pene- trability would lead to a curve whose rise was nearly complete after about 500 kev; even more striking, the cross section observed is about fifty times smaller than such similar reactions as Li 6 (p,a)He 3 . This points again to the absence of s-wave particles, which experience a much smaller barrier. We must assume that such particles cannot induce the reaction from the compound state of Be 8 involved. The reaction is governed by a strong selection rule: the conservation of parity. The two identical alphas emitted obey Bose statistics; they must have a wave function invariant under their exchange. But to exchange two alphas is equivalent to reflecting their wave function in the origin, since they have no spin. The alpha wave function must then have even parity. Their relative orbital motion will have a parity of (—1)', where I is the orbital angular momentum. We have then shown that the parity of the system is even in the final state. Since parity is conserved, it must also have been even in the initial state. But all simple nuclear models agree in giving the ground state of Li 7 odd parity. The incoming proton must therefore have had odd parity if the two alpha-particles are to be formed. This requires that the wave function for orbital motion of the incoming proton have an I such that $ ~ Yi m gives odd parity. The values I = 1, 3, ■ • • will do this; of these, the most easily penetrating partial wave is that with 1=1, the p-wave protons we observe. Thus we can account for the excitation 'See (HI 1) and(H12). Sec. 7A] Some Typical Nuclear Reactions 91 o 12 3 Proton energy (Mev) Pig. 9. Relative cross section as a function of incident proton energy for the reaction Li\p,a)a. eg E C3 6 bO S3 o +3 . o 4 ^5 13 0.25 0.5 Proton energy (Mev) 0.75 Fig. 10. Relative cross section as a function of incident proton energy, for the production of gamma-rays by the reaction Li 7 (p,p')Li 7 * in the region of the major sharp resonance at 0.44 Mev. 92 A Survey of Nuclear Reactions [Pt. VI curve and the low yield by assuming even parity for the compound state involved. The angular momentum of the compound state must be the resultant of the incoming proton spin, S = J^i an d the target nuclear J = %, combined with the orbital angular momentum 1=1. This means that the total angular momentum of the compound nucleus Jc must be Jc = 0, 1, 2, or 3. But, if the state is to emit alphas, which have no spin, the angular momentum must be even (from the Bose statistics) and the state is then described by the quantum numbers J = or 2, even parity. The angular distribution of the alphas with respect to the fixed direction of the incoming beam can be no more complicated than the partial wave of the incoming beam, and in fact the wave function will be of the form if/ a ~ a + bYi°. Since Bose statistics requires that the wave function of the outgoing alphas be even in cos 6, the most general form possible is a + b(E) cos 2 6. This fits the experiment reasonably well up to some hundreds of kilovolts. Lately the higher- energy angular distribution has shown (H8) terms of the kind cos 4 6, which imply that incoming /-wave particles must be considered. The general features of the discussion are unchanged. But what of Fig. 10? The very sharp resonances for the emission of a 17-Mev gamma-ray must be explained. The cross section for this reaction at resonance energy, with I the spin of the target nucleus, is 7rX 3 rj,r 7 27 + 1 <Ty(E r ) ~ ■ (r p + r T ) 2 2 and, since the measured value of <r y (E r ) is 7.2 millibarns, with a resonant energy of 440 kev and a measured total width of 12 kev (F9, F10), we can write the value of T y as ffyl abs r T ~ (2/+l),rX 2 -i assuming only that T y /V p « 1. This yields a gamma-ray width for the excited state of the compound nucleus Be 8 of about 30 ev. Ob- viously a strong selection rule must be in play to prevent the decay of the compound state by alpha-emission, which would be expected to have a width of hundreds of kev. The possibilities are two: (1) Only s-wave protons are captured to form this level, giving it angular mo- mentum Jc = 1 or 2, odd parity, or (2) the same p-wave protons which give the previous reaction are captured, giving states with even parity, but with spins J c = 1 or 3. In either case, whether the compound state in question has odd parity or odd J c , it cannot emit two alphas. To Sec. 7A] Some Typical Nuclear Reactions 93 distinguish between the alternative, as to fix the J c in the two alpha- emitting states, requires further information. This has been provided by a beautiful experiment (D3) in which accurate measurements were made of the angular distribution of the gamma-rays. The experiments fully confirmed the dispersion formula in the interference form, Eq. (71), by the observed variation of the angular distribution as the proton energy was varied through resonance. Forward-to-backward asymmetry proves that two levels are involved in the gamma-emission, with opposite parities. It is natural to identify the wide interfering state with some state of the (p,a) reaction, neces- sarily even, and then the narrow resonant state must be odd, formed by s-wave capture. This is consistent with the observed isotropy of the gamma-rays near the sharp resonance. Two lower states take part in the gamma-emission: the ground state, transitions to which yield the famous 17.6-Mev gamma-ray; and a state at about 3 Mev excitation, nearly 2 Mev wide. Both these lowest states decay by alpha-emission. The general rule about the zero spin of even-even ground states would lead to the conclusion that the ground state here has J = 0. Some experimental evidence for a lifetime of this state near 10 — u sec does not seem strong enough to upset the rule, and evidence from the beta- decay of Li 8 (that the two lowest Be 8 states differ in J) supports the rule. The near-equality of the gamma-ray transition rates from the resonant state to the two low states suggests that they do not differ by many units in J. It seems likely that the 3-Mev state has J — 2, parity even. Then the choice between the two possibilities J e = 1 and J c = 2 for the sharp state seems direct; we assign the resonant state the quantum numbers J c = 1, odd, and all the effects are described; both gamma-rays are electric dipole, thus fitting with the sizable width. This neat account seems disproved by observations (C9) of the elastic scattering of protons at the 440-kev energy. The presence of a scatter- ing resonance whose interference with the Coulomb scattering has been studied shows that at least most of the compound nuclei are formed by p-wave, and not by s-wave, capture. Then the isotropy of the resonant gamma-rays is in the nature of an accident, (C4) implying not that the protons brought in no evidence of direction, but that the compound state with J = 1 was formed with equal probability in the three sub- states J z = ±1, 0. This would give isotropy; it will take place, accord- ing to the rules for combination of angular momentum matrices, if the triplet proton wave (spin parallel to orbital angular momentum) is cap- tured about one-fifth as frequently as the quintet wave (D4). Then the interfering state is some non-resonant odd state, and the sharp resonant state must be assigned the description J c = 1, even. The 94 A Survey of Nuclear Reactions [Pt. VI gamma-rays are magnetic dipole and electric quadripole in a particular mixture. This is a less simple and less satisfying story, but it seems required by experiment. This account is a rather sketchy summary of one reaction, given as an example of the many-sidedness of the problem of nuclear reaction spectroscopy in the region where individual levels are of importance. The example at hand is not the simplest known case, but it is very far from the most complicated. The discussion of each reaction in great detail will be the content of a final nuclear spectroscopy. (6) Some Other Examples. Proton capture by C 12 and C 13 has been much studied. Here the fit to the one-level dispersion formula is excel- lent, though one must take into account the variations in T p coming from the appreciable change in penetrability over the 40-kev width of the resonance, which is some 2 Mev below the top of the barrier. It is worth while to note that the shape of the 440 kev resonance in the Li 7 (p,y) reaction does not fit the Breit-Wigner formula especially well. There is a superimposed asymmetry, with a rising gamma-ray yield on the high-energy side of the resonance. But in order to ascribe this to a variation in penetrability — arising from the capture of a proton partial wave with high I — a very high centrifugal barrier would have to be invoked (B19), and this would make the cross section absurdly small. Some more complex level scheme must be employed. The complicated and beautiful studies of the reactions arising from F 19 + p have led to repeated analysis from the point of view of the one- level formula (Fig. 11). Here the problem is to see if all the fifteen or more levels observed can be fitted into a scheme which gives reduced widths y r for the various reactions, each roughly independent of the level involved, and tries to account for the great experimental variation in width and strength of levels on the basis of penetrability changes due to* energy and angular momentum values alone. The situation is still not entirely unraveled, but the scheme is probably workable without too much arbitrariness (S2). There is increasing evidence of the value of another approximate constant of the motion, the so-called "isotopic spin" (W8). This quantity describes the spatial symmetry of the nuclear wave function under the interchange of any two nucleons. It can only be approxi- mately conserved, of course, since in fact the neutron and proton differ at least by Coulomb effects. Isotopic spin would be an accurate quan- tum number, and the nuclear wave function would consist of functions with one well-defined isotopic spin value, only if all nucleon-nucleon interactions were identical, independent of the type of nucleons involved. How strong this partial conservation law really is has not yet been estab- Sec. 7B] Some Typical Nuclear Reactions 95 lished. A similar situation exists for the spin, which can be intercon- verted with orbital angular momentum only in the presence of non- central forces. Strong forces of this type exist, and hence spin is in general not even approximately conserved; only total angular momen- tum, a strict quantum number, remains as a good basis for angular momentum selection rules. Whether the spectroscopy of nuclear levels based on the dispersion theory can be carried beyond these light nuclei is dubious. Even here A i ks^-J 0.4 0.8 Proton energy (Mev) Fig. 11. Relative gamma-ray yield (schematized) as a function of incident proton energy for the reactions coming from F 19 + p in the region below 1 Mev incident energy. Note the many and varied resonance peaks. the level schemes which have been proposed are highly complex. With- out an a 'priori guide to the position and character of nuclear levels, the job of spectrum analysis is difficult. It will be noticed that the problem of the light nuclear reactions is quite parallel to the problem of the low- lying states revealed in the gamma-decay schemes of radioactive nuclei. In the present example, however, the natural widths of levels is directly measured, while the number of possible reaction products adds the complicating factor of competition. But there seems little reason to doubt that the general scheme proposed is adequate. B. Reactions without Marked Resonance We have seen in Section 5 how the statistical theory of reactions proceeds by looking apart from the details of individual levels, and in Section 7A we have discussed in detail some reaction types which are clearly suited to the description of dispersion theory, in which the proper- 96 A Survey of Nuclear Reactions [Pt. VI ties of single levels are made the basis for understanding. Here we shall apply the ideas of the statistical theory to the large class of reactions in which no marked properties of levels are observed; the theory is intrinsically suited to such reactions. We remark first that these reac- tions are generally characterized by high excitation of the compound nucleus. Wherever this occurs, the level spacing decreases and widths increase, as we have seen, so that either for all circumstances or for the conditions of nearly all practical experiments the importance of indi- vidual resonances is negligible. Excitation of nuclei by neutrons means the formation of a compound nucleus with energy of excitation equal to the neutron binding energy at least. As shown in Section 7A, this region (say 8 Mev in nuclei from A ~ 100 to about 200, and perhaps 6 Mev or less thereafter) is marked by discrete levels, spaced from a few hundred to a few electron volts, when the incoming neutron has negligible kinetic energy. As the kinetic energy of the neutron beam increases through the region of marked resonances, the level density increases rapidly. At a few kilo- volts at most, the present technique does not permit resolution of indi- vidual levels. We have already shown [e.g., in (72)] that the average over many levels of the dispersion formula leads to a result agreeing with the statistical theory. It is of interest to apply this idea to the actual case of the absorption of neutrons with energy in the range from a few to a few thousand electron volts, with the middle and heavy nuclei as targets. If we stick to s-wave neutrons only, we can write for the average absorption cross section, averaged over many levels, with level density p (0) = 2xp ■ xX 2 — — ; — — (bars indicate average value (79) over manv levels') J- n± a * ' a over many levels) [Compare (71).] From the expansion for f(E) given in (60) we can write r„ = — 2kR/8, with 8 and R constants which can be given a rough interpretation as determining properties of the function z{E) [see (62)ff.] or can be regarded as empirically determined. We find that r a y> F„ for energies below a few kilovolts, and then in this region we would expect -co, ^ R !^ " a ~ k ' s where R, p, a, 8 can be roughly estimated. Note that this applies to averages over many levels, not to thermal cross sections. Although there is no direct check yet on this prediction, it seems consistent with Sec. 7B] Some Typical Nuclear Reactions 97 the general knowledge of absorption cross sections. A more detailed extension of this similar theory has given results 1 which are not wholly in accord with the data, at higher energies especially, but which seem to demonstrate the essential correctness of the approach, granting its rather high degree of arbitrariness in fitting the results of single experiments. The scattering cross section can be treated in the same way. Here of course the potential scattering term e* s sin S is important. Even for very simple assumptions about level widths, a rather complicated result is obtained which is not yet wholly confirmed. Both at thermal and at higher energies the importance of the "impenetrable sphere" effects of the potential term are evident in the total cross sections where no strong absorption resonance is present. Variations of radius R as well as special structural features of some nuclei seem to be present in the detailed comparison of theory with experiment. 1 Agreement is much better for some heavy and for some lighter nuclei; a few middle-weight nuclei do not give even the expected energy dependence of <r„ (0) , i.e., 1/E* A . 1. The Statistical Approach. For sufficiently high energy of the bom- barding particle, no resonances will be observed. This can arise out of the experimental conditions : it is hard to define the beam to a very small energy range as its mean energy grows, and even the temperature motion of the target nuclei will produce a comparatively large uncertainty in relative energy for high bombardment energies, as we can see from relation (74). More significant, however, is the fact that the levels themselves become broader as the energy of excitation increases. This is the obvious consequence of the fact that higher excitation energy quite generally makes available many more modes of decay from a given level. In this region of high energy, then, the dispersion theory will have little value, for many levels, each with its unknown but specific properties, will take part in every reaction. In the preceding section we showed how the dispersion theory treat- ment went smoothly over into an entirely statistical form when the level spacing was small compared to the level width, for the particular case of neutron absorption. We replaced the precise values of the level widths by averages over many levels; such averages ought to vary smoothly with energy. In a sense we here calculate not the result of a particular reaction with a definite target nucleus, but a kind of average over many nuclear species very close in A and Z to the actual target. This ought to give then the general course of any reaction, fluctuations about the average behavior which will show up in particular cases being disre- 1 ThisTvhole treatment follows closely that of (F7). 98 A Survey of Nuclear Reactions [Pt. VI garded. The recent neutron scattering work of the Wisconsin group seems to show such fluctuations (B3). The fundamentals of the statistical theory were given in Section 4; here we propose to apply them. The main relations are two in number. The first, which is given in Eqs. (40) and (44), is the statement of the main Bohr idea of the nuclear reaction progressing in two independent steps: <r iiP = Sri ■ JS?- = xX* 2 E W + DPiiia ■ ■=£ = *i(fi) ■ ~~ (80) Here we have introduced the cross section <ri(c) for formation of the compound nucleus with incident particle i. The second is the statistical relation between level width for a given disintegration and cross section for the process inverse to the disintegration [see (33) and (34a)] : r ia (W) Pc (W) = ^ ~ (81a) and Y nl = kGP n ,i (81b) From these two we can find either the width or the cross section, using basic ideas, and compute the other from their relations. (a) Neutron Reactions. Here we have only the centrifugal barrier opposing the contact of neutron and target nucleus. Let us consider first the case of high-energy neutrons, with energies so high that the wavelengths corresponding are small compared to nuclear dimensions. In such a case we may follow the classical trajectory of the incoming particles. They will form the compound nucleus with sticking probabil- ity whenever they strike the nuclear disk. Moreover, it is evident from our whole nuclear model that for energies above, say, a few Mev the value of in is essentially unity: every particle that touches the nuclear matter sticks. Now we may write the contact cross section for neutrons in the partial wave of orbital angular momentum I as we did in Eq. (49) : tr n ,l = (21 + l)*K 2 Pn,itn.i (82) But for high energies we set £ n i -* 1, and the quantities P n ,i [found in (36)] take the values P n ,i — > 1 for low I's, such that the centrifugal barrier is well below the energy available, say for I below l c , the critical angular momentum. For I > l c , the values P„,; rapidly approach zero, and, if many Z's are involved, we can neglect the transitional cases near l c . Sec. 7B] Some Typical Nuclear Reactions 99 Then just as in (44b) the cross section <r n (c) for formation of the com- pound can be written C n (c) = 2(TiZ l c ~hR = ttX 2 23 (2* + l)P,m -» ttX 2 E (2Z + l)P Bl! 2=0 o ^irP 2 (high energy) (83a) which is of course the classical value. At low energies, we can give no a 'priori guess about the quantity £„,i. But relations (81) for level width can be applied. We write Tm = kyP ntl = ^- . f=I = i*- (2Z + l)P n fa (84) 2:rp irX 2irp It now seems inviting to make the identification : 2irp7 £„,* = * (85) gn Here the functions p and 7 depend only on the compound nucleus [while gR is the statistical factor needed in <r n i, by (69)], and will vary very little for small changes in neutron kinetic energy. We write this: k £» = - and k <r n (c) = ttX 2 2(2Z + l)P ni! • - (low energy) . (83b) K. with the constant K to include the unspecified functions above. The result is then a sticking probability which is proportional to k for slow neutrons, which can be justified from the side of the dispersion-theoretic discussion in Section 5. As energy increases, the sticking probability goes over to 1 gradually, and we may choose the constant K to give a smooth extrapolation from low- to high-energy values, from (83a) to (83b) . The resulting neutron cross section behaves like c n = icX 2 -P ,i-{k/K)~l/v at the lowest energies, where only the s wave, with I = 0, contributes and gradually falls with increasing energy until it approaches the geo- metric cross section. It is to be remembered that all this refers to the cross section for actual formation of a compound nucleus. The "potential scattering" term e id sin 5 of (68) is present in addition. Part of the compound nucleus 100 A Survey of Nuclear Reactions [Pt. VI formation cross section may indeed lead to re-emission of a neutron with just the incident energy, a process of elastic scattering after com- pound nucleus formation. 1 This part can in fact give coherent inter- fering contributions to the diffraction effects. Such interference terms can complicate the shadow scattering. In principle we would expect angular distributions of elastically scattered neutrons to show a more or less isotropic part, coming from the re-emission of absorbed neutrons from several overlapping levels in the compound nucleus, together with a more complex and usually well-collimated part, which would contain the diffracted shadow waves plus interference terms from both sources. In the statistical theory, however, we expect the part of the cross section of compound formation we discuss to approach the value wR 2 for high energies, though the total cross section, including the elastic diffraction or shadow scattering, will become very nearly twice that. (b) CJiarged Particle Reactions. Here the Coulomb barrier introduces an added complication. We begin with the last member of (80) : <n(e) = 7rXi 2 2(2Z + 1)P„{« (80) It is now in the spirit of our statistical approach to give the sticking probability the same form for charged particles, once they have reached the nuclear surface, as we found it to have for neutrons, for the nuclear forces are predominant there. If now we write the W.K.B. expression for the penetrability P v j from Eq. (37), we have, in the region E/B « 1, where the W.K.B. method is reliable: <r,-(c) = ttX 2 2(2Z + l)fc, • (^) V 2c < = ,rX 2 2(2Z + l) e " 2ci T^?~ Now the constant K can be chosen to make the cross section for the low- energy, high-barrier region, where the W.K.B. expression is adequate, go smoothly over to the high-energy limit, which is just aiic) = ttX 2 2(2Z + l)P n ,i -» x£ 2 with the same arguments about the critical angular momentum l c as in the case of the neutron. We finally obtain then, for incident charged particles, i, d{c) = xX 2 £ (21 + l)e- 2Cl for all energies (86) 1=0 A long literature exists on the points we have treated here quite heuris- tically. By and large, the simplest justification for our treatment is found in the related discussion of Section 5, but compare also the papers 1 Cf. discussion following Eq. (45). Sec. 7B] Some Typical Nuclear Reactions 101 of Konopinski and Bethe (K8) and Bethe (B13), whose results we have essentially set out above, but by rather different methods. We include here a number of graphs (Figs. 12 to 14) showing the re- sults of computations of the cross section for compound nucleus forma- tion, based on (83) and (86), and using the same value both for the target radius {R — I A- A ^-10 -13 cm) and for the radius of the projectile. X -^proton Fig. 12a. Cross sections for protons, for formation of the compound nucleus, plotted linearly. The graphs apply to protons, alphas, and neutrons over a wide range of energies, for typical target elements over the upper two-thirds of the periodic table. Following these graphs, we include graphs (Fig. 15) of the specific widths, the functions defined in (50b). These graphs are based on the cross sections given in (83) and (86), and on the as- sumption concerning the statistical level density of compound nuclei, given in (31). From the specific width graphs we may compute the function D p = I^/STy = /p/S/y for any definite reaction, and hence the total reaction cross section on the statistical theory. The special problems raised by the deuteron and the gamma-ray will be discussed later. The specific widths are given for typical nuclei as functions of the energy of excitation of the compound nucleus, for the several possible emitted particles. 102 A Survey of Nuclear Reactions [Pt. VI 0.2 0.4 0.6 0.8 X = "proton /^barrier 1.0 Fig. 12b. Logarithmic plot of proton cross sections. Fig. 12c. The proton barrier height needed for the cross-section graphs, as a function of Z. The parameter x is given in terms of proton kinetic energy measured in the laboratory frame. Several values for R have been assumed. In the cross-section curves we use R = 1.4(A M + 1)10 -13 cm. x-E a /E barrier Fig. 13a. Cross sections for alpha-particles, for formation of the compound nucleus, plotted linearly. 103 104 A Survey of Nuclear Reactions [Pt. VI a! 0.5 0.6 0.7 0.8 0.9 1.0 Fig. 13b. Logarithmic plot of alpha cross sections, for energies below the barrier. x -E^/E h!Lrrier 15 > .SP 10 5 - 10 20 30 40 50 Z 60 70 80 90 100 Fig. 13c. The alpha-barrier height needed for the cross-section graphs, as a function of Z. The parameter x is given in terms of alpha-particle kinetic energy measured in the laboratory frame. (Lowest barrier has been used in cross-section graphs.) 10 \\ ° £ 5 b 1 -— — ° o experimental points for Ag £=90 O Z=GQ 1 1 Z=30 i i i i i 123456789 tf„(Mev) Fig. 14. Cross sections for neutron absorption in three elements. 105 10 106 A Survey of Nuclear Reactions [Pt. VI The same calculations which lead to the specific widths give (before integration) the energy spectrum of emitted particles. We have already shown a typical set of spectra in Fig. 6. 11 13 15 9 R (Mev) Specific widths for neutron, proton, alpha-particle, and gamma-ray emis- 7 E, Fig. 15. sion from several compound nuclei, as a function of energy of the emitted particle. The specific widths are computed for residual nuclei of even A and Z. For odd-even or even-odd residual nuclei, multiply the plotted width by 2; for odd-odd residual nuclei, multiply plotted width by 4. Interpolation for other Z can be made directly. The application of these formulas to all the reaction types listed, in the appropriate energy region, is straightforward. To point out a few of the more general consequences, and to provide a model for calculation, Sec. 7B] Some Typical Nuclear Reactions 107 we shall discuss one or two particular reactions which can be compared with experiment. (c) Competition. The reactions (a,ri) and (a,2n) on Ag 109 have been studied by Bradt and Tendam (B20, Gl). Figure 16 is a simplified version of their experimental results, and of some later work on similar 300 O In(oi, n) 7 Ag i? = 1.4-(A 1/s + 4Vs). NT 18 cm Ag(a,«)-A ; §200 0) > §100 Sn i? = 1.5-^ 1/a -10- 13 cm 14 15 16 E (Mev) Fig. 16. Comparison of experimental and theoretical cross sections. The circles refer to the measured relative cross section for In(<*,n) (see reference Gl), and the triangles to the work of (B20) for Ag(a,n). Two theoretical curves are fitted: one with R = 1.5-A^-lO- 13 cm, and one with R = 1.4 -(A* + 4«)-10- 13 cm. The difference is a fair indication of the limit of reliability of our statistical theory as well as of the assumption R ~ A^. reactions. The cross section for formation of the compound nucleus, with sticking probability near unity for these energies, is plotted as well (the graphs of Figs. 12 to 14 were used). The compound nucleus can emit a neutron, two neutrons, charged particles, or gamma-rays. The nucleus is excited by the energy E a + B a . The binding energy of the alpha, B a , is not known, but can be written B a = — 4B„ + 28 Mev, where B n is the average binding energy of each of the nucleons brought in, and the mass defect of the alpha is known to be 28 Mev. We cannot give B n with any accuracy in this region of the table, but it is pretty surely between, say, 4 and 8 Mev. The compound nucleus is then 108 A Survey of Nuclear Reactions [Pt. VI excited to between 15 and 20 Mev in these experiments. Such an excita- tion energy brings us well into the region of dense levels. The escape of a gamma-ray will be negligible here, for, as we shall see, the gamma widths are always small compared to heavy-particle widths when the excitation is appreciable. Charged particle widths will also be small compared to neutron widths, even though the excitation energy is a good deal beyond the barrier for protons, because, from expression (52) the release of a proton with the full available energy is not probable. It is probable that the emitted particles take off energy corresponding to the nuclear temperature, here some 2 Mev, and for such protons the barrier is still effective. We expect that neutrons will go off almost all the time: T n ^> T p . The results of experiment agree nicely with this idea from the lowest energies measured up to about 15.5 Mev. Here the rate of increase of the cross section for (a,n) drops sharply. With alpha-particle energy of 4 Mev higher, the value of tr a , n has fallen to a third or less of the value of <r a (c). This is the typical effect of competition. The factor D n has changed from very near unity to something much smaller because the threshold for a new reaction has been reached at 15.5 Mev. The new reaction is the (a,2n) reaction, which leaves the residual excited nucleus In 111 . We may estimate the probability of this reaction very roughly as follows: The cross section for formation of the compound nucleus is <r a (c), given in Fig. 13; £ is about unity; and the fractional number of neutrons emitted with energy E n is, from (52), just — — = E n <r n e E " /T (87) In order to emit a second neutron, the first neutron must leave the residual nucleus sufficiently excited to emit the second one. This will occur whenever the residual nucleus has even slightly more energy (by a few kev) than the binding energy B n of the neutron in the residual nucleus, for the gamma-width is then negligible, and charged particle widths are all extremely small because of the Coulomb barrier. We can then write In(En) dE n a(a,2n) = a a (88) " En, max I n (E n ) dE n where the integral in the numerator is carried from zero removed energy to AE = E n max — r 2n , with V 2 „ the threshold energy for emission of a second neutron. Both integrals can be approximated by using the Sec. 7B] Some Typical Nuclear Reactions 109 Maxwell distribution of (87) : 1 - (1 + AE/ T )e- AE/T <r{a,2n) = <r a (89) 1 (i + ^=). max/T where AE = i? max — r 2 „ is the excitation energy surplus beyond the threshold of the 2n reaction, i? max = E a + B a — B n . The threshold for the observed reaction in silver is 15.5 Mev, and E m&x ~ 2 or ZAE, so 21- OJ 1 £1 Bi 209 i2 = 1.4-(A 1/3 + 4 1/3 )-10" 13 Ni 60 ■ a m i? = 1.3-U 1 /3 + 4 1 /3).i -ii^^ / ,. (a,2n) (a,w)+(a,2w) □ (<*.n) 10 20 30 40 E a (Mev) Fig. 17. Comparison of theoretical contact cross sections with the sum of observed (a,n) and (a,2n) cross sections. The solid curves are the theoretical values and are marked with the radii assumed. Ni 60 data from (G2); Bi 209 data from (K5). we can set the denominator equal to 1, and the fit with theory is then excellent up to 18 or 19 Mev, as far as the experiments go. Agreement was obtained by choosing the best value of the temperature r, which turned out to be the very reasonable value 1.8 Mev. Our simple model (Table 4) gives this value for silver excited about 18 Mev, which is not far from what we would expect for the excitation of the residual nucleus to the limit of the neutron spectrum, with alphas of about 15 Mev. Other multiple reactions, as (n,2n), and even the more highly multiple reactions, may be treated in a similar fashion. The literature contains discussions of the most characteristic features of many other reaction types (see Appendix II). We include more experimental comparisons in Fig. 17; the entry of successive competing reactions is beautifully shown (G2, K5). 110 A Survey of Nuclear Reactions [Pt. VI SECTION 8. THE DEUTERON AS A PROJECTILE We have not so far discussed what is the most commonly used nuclear projectile excluding pile neutrons. This is the deuteron, 1H 2 . It is essentially different from the proton and the alpha-particle, because neither is it a simple nucleon, nor is it so tightly bound, like the alpha, that its state of lowest internal energy alone plays a part in nuclear reactions at ordinary energies. On the contrary, the deuteron is so loosely bound — only (2.23)/2 Mev per nucleon — and so large a structure, with its constituent nucleons often 3 or 4 • 10 -13 cm apart, that these properties play the major part in determining the course of deuter on- induced reactions. The use of deuterons to produce nuclear reactions, especially for the copious production of radio isotopes, is by now traditional. An excellent set of data (C7) has been published which gives the practical yields — often quite large — from deuteron bombardment at moderate energy. We give a few typical numbers in Table 7. Perhaps the most evident TABLE 7 elected Experimental Yields with 14-Mev Detjt (Thick-Tabget) Yield Product (nuclei/10 6 Reaction Isotope deuterons) Mg 24 (d;a) nNa 22 500 Na 23 (d;p) nNa 24 1400 AF(d;p,«) u Na 24 5.7 p 3I W;p) 15P 32 860 Cr 52 (d;2n) 2S Mn S! 140 Cu 63 (d;p) 29CU 64 270 Br 81 (d;p) 36 Br 82 220 Te 130 (d;2n) 63l 130 95 Te 130 (d;n) 63 I 131 180 consequence of deuteron bombardment is the high excitation of the compound nucleus formed by deuteron capture when it takes place. The nucleus is excited by the kinetic energy of the beam plus the full binding energy of a proton and a neutron, reduced only by the weak 2-Mev binding of the free deuteron. This amounts in general to some 14 Mev plus the kinetic energy. Such an excitation implies the validity of the statistical theory. The penetrability may be calculated as in (37) and in Figs. 12 to 14 in a fair approximation to the deuteron contact cross section, for a deuteron of energy Ed on a nucleus of charge Z is Sec. 8A] The Deuteron as a Projectile 111 that for an alpha-particle with energy (2^E d ), but with a target nucleus of charge Z/2 Vi , i.e., an alpha with E' = \AE d , Z' = 0.6Z. (This implies the rather rough approximation, A proportional to Z.) Such deuteron-induced reactions would lead nearly always to neutron emission, and often to multiple-particle emission. The neutron can take off sizable energies because of the high nuclear temperatures. It is nevertheless striking that many reactions have been observed, espe- cially with the heavier nuclei, in which proton emission was of com- parable frequency with neutron emission. This is the case not only at higher energies, where the proton barrier might be unimportant for the nuclear temperature involved, but even at moderate energies, and espe- cially in the heaviest nuclei. We should like to discuss this phenomenon in detail; it is the result of break-up of the deuteron, and it is called stripping. A. The Oppenheimer-Phillips Reaction : Low-Energy Stripping The deuteron moves toward the nucleus in the presence of the large electrostatic field produced by the nuclear Coulomb charge. The proton in the deuteron is repelled from the nuclear surface, but the neutron feels no such force. For the highest energies, or for light nuclei, when the Coulomb barrier is unimportant, the deuteron is polarized by the Cou- lomb field, but it may not break up. In cases where the Coulomb barrier is important — say comparable to the binding energy / of the deuteron — the proton will not reach the nuclear surface when the neutron does. The Coulomb forces will repel the proton, and the nuclear forces will seize the neutron. The compound nucleus then will be formed not by the capture of a deuteron, but only by that of a neutron, and the proton, which will never have come within the range of the nuclear forces at all, will fly off with a gain in kinetic energy, both from the recoil of the broken deuteron bond, and from the Coulomb repulsion. The ratio of the barrier height to the deuteron binding energy will be a very rough measure of the importance of the process. This process is sometimes called, after the first authors to recognize it, the Oppenheimer-Phillips process (01). The process can be treated by dividing it as usual into several steps. We write the cross section for a reaction initiated by a deuteron of energy E d resulting in the release of a proton of energy E p in the interval dE p as follows : a(E d ,E p ) dE p = <r d (po,R)lT(E p ) dE p (90) Here a d is the contact cross section for penetration of the proton com- ponent of the deuteron, not to the nuclear surface, but to the more or 112 A Survey of Nuclear Reactions [Pt. VI less well-defined distance p to which the proton has the maximum probability to penetrate; T(E P ), called the transfer factor, takes into account the gain in energy of the proton as it moves out, and £ is the sticking probability of the neuteron averaged over all available levels into which it may be captured. The modified penetrability function for the deuteron, ad(p R), taking into account the polarization produced by the Coulomb field, has been given, with useful graphs (VI). The factor £ will increase with energy of excitation, perhaps as rapidly as the level density, and will become unity for high excitation energy, which means low proton energy. High proton energy will mean low excitation energy, and may indicate the existence of individual levels. For low deuteron energies, the excitation of the nucleus will be quite low, even less than that following slow neutron capture, since the binding energy of the deuteron will be lost to the compound nucleus. The neutron sticking probability can be quite low at such "negative kinetic energy" but, even then, the easy escape of the proton from its rather distant point of closest approach will result in a large Oppenheimer-Phillips cross section and a small deuteron capture. This cross section will be larger than deuteron capture just for this reason, even when the neutron does not always stick. For higher deuteron energies, the proton can approach the nuclear surface rather closely, and the most probable distance, p , will not be much greater than R. Then the ordinary deuteron capture and neutron emission will have an excitation function much like that of the Oppen- heimer-Phillips (d,p) process. But, by reason of the high excitation following capture of a deuteron, the (d,p) process dominates the (d,n) process up to energies well above the barrier, since the (d,pri) process is very likely to occur after deuteron capture. B. Stripping Reactions at Higher Energy Even at very high energies, of course, the passing deuteron is dis- torted by the nuclear Coulomb field, and the Oppenheimer-Phillips process takes place. Here one may think not so much of the purely electrostatic forces as of the passing deuteron in a varying field due to the nucleus. In a frame of reference in which the deuteron is at rest, and the heavy nucleus is moving almost undeflected past it, the electric field experienced by the deuteron is a highly transverse field, owing to the Lorentz contraction of the nuclear force lines. This pulse of electric field can be Fourier-analyzed into a collection of nearly transverse waves of a wide spread of frequencies. These waves behave nearly like quanta, and are often called "virtual quanta" in this method of calculation (Dl). The virtual quanta may induce photodisintegration Sec. 8B] The Deuteron as a Projectile 113 of the deuteron as they pass by. In the laboratory system, where the deuteron is moving, the process becomes an electromagnetically induced stripping. It is in general a minor contributor to deuteron reactions compared to specifically nuclear effects at high energy. But, much more important in the high-energy limit, where the deu- teron may be handled classically, with a well-defined trajectory, is a process we may call collision stripping (S7). Here the deuteron is broken up by actual contact of one of its particles with the nuclear matter, while the other particle flies off with the momentum it had at the instant of collision, the resultant of its share of the deuteron center- of-mass motion and of the internal motion within the deuteron. This process yields neutrons and protons in equal numbers, and in a cone restricted to the forward direction, with a half-angle given by the simple relation 6 ~ (W d /E d ) V2 , and energies given by E p ~ E n ~\(E d V2 ± Wd Vi ) 2 , where W d is the deuteron binding energy. The process is evidently one which leads to different energy and angle distribution from the low-energy Oppenheimer-Phillips reaction, or from emission after formation of a compound nucleus. Although detailed agreement with this simplified form of the theory is to be expected — and has been veri- fied — only for really high-energy deuterons, at 190 Mev, the process must contribute continuously at lower energy, and finally merge with the standard Oppenheimer-Phillips case, in which the "stripping" occurs with by no means negligible reaction between the captured nucleon and the one that goes free, and in which the trajectory is spread heavily by diffraction (P2). With deuterons in the region of moderate energy, say from 5 to 20 Mev, the stripping process is still dominant. Forward-peaked and even more complicated angular distributions of neutrons from these projec- tiles on a variety of targets of nearly any range of A indicate that neither compound nucleus formation nor a statistically treated stripping, in which classical ideas are used and all orbital angular momenta are regarded as contributing, can completely account for the process (B23). With good energy definition of the incident beam, it is possible to fix upon a group of outgoing protons of a definite energy. These protons arise from the capture of the neutron into one specific level of the prod- uct nucleus. At these moderate energies the excitation of the nucleus may be less even than that following capture of a free neutron, because the deuteron binding energy must be supplied to free the outgoing pro- ton. This implies that individual levels will separately contribute. Since the incident deuteron momentum in the stripping approximation point of view is just equal to the sum of the momentum of outgoing proton and of the neutron before capture, forward-emitted protons may 114 A Survey of Nuclear Reactions [Pt. VI carry most of the available momentum. The internal deuteron motion does not often permit large values of momentum for both nucleons. Thus the captured neutrons frequently have long wavelengths, and quantum effects become decisive. The loose deuteron structure means, moreover, that large values of the orbital angular momentum of neu- tron with respect to nucleus can be important, while only a few I values can be captured into a single level. Then penetration effects may show up strongly in the angular distribution of the stripped protons, espe- cially at small angles. Study of the fine structure of these distributions determines the partial waves captured, and becomes a powerful means of level spectroscopy (B22). The really complicated behavior of the deuteron as a projectile, which arises from its loose structure and internal motion, seems qualitatively explained by the various processes outlined; it is by no means clear in full detail. SECTION 9. RADIATIVE PROCESSES IN NUCLEAR REACTIONS The all-important role which radiation plays in the de-excitation of atomic states is diminished in nuclear reactions by the numerous alter- native means of decay. In nearly all reactions, nevertheless, gamma- rays are observable products, since particle emission cannot always be expected to leave the nucleus in the ground state, and, once the cooling nucleus has dropped below the energy content representing the threshold for particle emission, only radiation can take away the remaining energy. With the new machine sources of continuous gamma-spectra, moreover, nuclear reactions induced by gamma-rays are of high interest. The fact that the electromagnetic interaction between field and charge- current is completely known makes the study of nuclear structure through radiation, rather than particle collision, seem attractive. The somewhat illusory nature of this argument comes both from the ex- perimental difficulty of accurate measurement for processes of generally low yield under conditions of poor energy resolution and from the com- plex character of the charge-current vector within nuclei, which turns out to depend on rather fine detail of nucleon motion, as on the presence of exchange forces and other phenomena connected with the intranuclear motion of the meson cloud. But the whole subject is full of interest. A. The Multipole Classification Just as the emitted particles are classified by the orbital angular momentum of their partial waves, which determines angular correla- Sec. 9A] Radiative Processes in Nuclear Reactions 115 tions and penetrabilities, so it is useful to classify radiation from any charge-current system contained in a limited volume of radius R in a similar way, according to the successive terms of a general expansion. This expansion is called the multipole representation, and it amounts to a sorting by angular momentum and parity. The electromagnetic field is a vector field, with some special properties due to the zero mass of the particles of the field — photons — and it turns out that there are two independent partial waves (analogous to the familiar two types of polarization) for each value of total angular momentum radiated away. Speaking physically, one would expect three possible angular momentum values for each orbital value, since the photon has unit spin. But the condition of transversality, which is closely related to the zero mass of the photon, excludes one orientation. In Table 8 are presented the TABLE 8 Multipole for Given Transition Parity Radiated Angular Momentum, J y Change 1 2 3 L (even) L (odd) Yes No None None Elec. dipole Mag. dipole Mag. quad- ripole Elec. quad- ripole Elec. octo- pole Mag. octo- pole Mag. 2 L -pole Elec. 2^016 Elec. 2 i -pole Mag. 2^016 selection rules and the type and order of multipole for a given parity change and radiated angular momentum. The radiated angular mo- mentum is of course to be taken in the usual sense of the vector model : if Ji and J/ are initial and final angular momenta, the relation J t - + J y > Jf > | Ji — J y | must be fulfilled. In general only the lowest order of multipole allowed by the selection rules contributes; e.g., if the transition is Ji = 2, odd —►«// = 1, even, we expect electric dipole only. The magnetic multipole of order L is reduced in intensity with respect to the electric dipole of the same order in the ratio (h/McR) 2 . For electric and magnetic dipole this is easy to see by simply writing the ratio of the familiar dipole moments: elec. dipole moment eR mag. dipole moment eh/Mc 116 A Survey of Nuclear Reactions [Pt. VI It holds more generally. The absence of any J y = forbids zero-zero transitions completely, with one quantum emitted. For a transition in which a magnetic multipole fulfills the selection rules with the lowest value of multipole order L, the next higher electric multipole may con- tribute radiation less by an order of magnitude, since the reduction from order to order in L is estimated to be in the ratio (R/\) 2 < (p/Mc) 2 ~ (h/McR) 2 . If the lowest contributing multipole is of elec- tric type, the magnetic multipoles are negligible. 1. Reduction of the Nuclear Dipole Moment. Classically, and even in atoms, the electric dipole transition is the most intense for radiation not too small in wavelength compared to the dimensions of the system. From our selection rules we could expect to see forbidden transitions, slow and non-dipole in character whenever large angular momenta have to be radiated. This is of course the origin of the well-known nuclear isomers, and of certain forbidden lines in the nebular atomic spectra. But there is a general argument which indicates that electric dipole transitions will not be so important, at least for nuclear gamma-rays up to some 10 Mev. The interaction, energy between a system of charges and the electro- magnetic field may be written : #int = 2] A-ji = X) A-(pVi) i i In the usual dipole approximation, we replace the operator v* by pi/m, and use the familiar relation between matrix operators, p = m di/dt. Then the dipole moment operator becomes Se t pi D = SejTi cc mi But plainly it is only the net displacement of the charges with respect to the center of mass which produces radiation from an isolated system. Then the operator whose matrix element between initial and final states determines the rate of radiation is P= X. — (X.--X) with AMX = MX> p + MX)x„ all particles TYli p n where M is the nucleon mass, for a nucleus with A nucleons, Z protons, and therefore N = A — Z neutrons. Here x p is the vector position of the pth proton, x„ that of the nth neutron, and X is the coordinate Sec. 9A] Radiative Processes in Nuclear Reactions 117 vector of the center of mass, all in an arbitrary reference system. Now we can write 1 M ! ( i -D? x "- e f? x " MA„ MA n (91) This is as though we calculated the dipole moment considering each nucleon with actual charge e* to have an effective charge only e,- — Ze/A. Clearly, if all the particles of the nucleus had e t -/m t - = e/M, there would be no electric dipole radiation whatever; the charge center and mass center would coincide, and the mass center clearly could not oscillate. In less extreme cases, where the motion of protons and neutrons is very similar — if they stick, say, more or less tightly together in alpha-particle sub-units — the electric dipole transitions would be at least much dimin- ished. There is no such general restriction upon the other multipoles. Indeed, magnetic multipoles will contain contributions both from the intrinsic magnetic moments of the nucleons and from other magnetiza- tions arising out of the transient currents coming from meson flow within the nucleus. These can be shown to be an inescapable conse- quence of exchange forces of various kinds (SI). In the deuteron photoeffect these currents are probably observable at low energy, where the major contribution is from a magnetic dipole transition. In general, then, we expect the magnetic multipoles to give somewhat larger con- tributions than the estimate made earlier would indicate, and we look for the electric dipole term to be much reduced, perhaps so much that the electric quadripole and magnetic dipole will overshadow its effects. 2. The Sum Rules. This reduction of the dipole moment is limited by a very general result, an extension of the Thomas-Reiche-Kuhn sum rule long used in the atomic case. We can write for the integrated dipole absorption cross section r x 2^ 2 e 2 h , I <T a (Ey)dE y = — — — 2^ f on J Mc „ Now, independently of how the so-called oscillator strengths, /„„ = | X on fK n | 2 , vary with the energy difference between ground and ex- cited state, E n — E = h 2 /2MX „ 2 , the value of the sum is just M mmj/ei eA 2 2-, Jon = 2-, — T~ \ J Wltn M any particle mass m; „ Sm,- i,j 2e \nii nij/ (92) provided only that the system absorbing the radiation consists of par- ticles of charge e,-, mass m,-, and that the forces between them are ordi- 118 A Survey of Nuclear Reactions [Pt. VI nary forces. Exchange forces contribute a correction of similar size. Applying (92) to the nuclear case yields the result /•" 27rVh 1 / _ e 2 \ 7T 2 / h \ 2 , NZ I <r d (E y ) dE y = ( 2 23 — o I = ( ) m p<? J c A¥ p V t£2e 2 / 137 \M p cJ P A NZ = 0.058 Mev-barn (93) A From (92) it follows that a nucleus with N = Z, consisting of a set of infinitely well-bound alpha-particles, would have a strictly vanishing / sum. This is the result the previous section predicted. But, since actually we take the nucleus to contain not fundamental alpha-particle building blocks, but protons and neutrons, perhaps bound with finite forces into alpha-particle-like units, the integrated cross, section will be given by the form of (93). Taking the two results together, we can reconcile them by observing that the cross section will remain low, and we shall have little or no contribution to the / sum, as long as the energy of the gamma-ray is insufficient to break up any strong correlations into alpha-like structures. But the dipole transitions cannot really be prevented by any such internal binding of finite strength; they can merely be deferred. Sooner or later, as energy increases, transitions are made to states lying high enough so that any given sub-unit is broken up, and the / sum begins to grow, reaching finally the total given by (93), which any system of protons and neutrons must eventually show, whatever their internal motion. (We exclude exchange forces for the moment; they change nothing qualitatively.) From these general considerations we would expect that the gamma- ray transitions from nuclei would be electric dipole, magnetic dipole, or electric quadripole in most cases, with the last two having possibly somewhat higher probability, for transitions involving only a few Mev (Wl). But, for higher-energy gamma-absorption or -emission, say from 15 to 20 Mev — energies large enough to excite any transient con- figuration of nucleons, even the stable alpha-particle — the dipole transi- tions begin to show their deferred dominance and lead to integrated dipole absorption cross sections of the order of an Mev-barn (D5), for nuclei of middle A. This indeed seems to be a fair picture of what is still a murky experimental domain. Further extension of the sum rule type of calculation leads to rough information on the values of quantities like I a y {E y )E y n dE y ; these, taken with the experimental data (HI, K3), tend to confirm the picture of the last paragraphs at least roughly (L4). Sec. 9B] Radiative Processes in Nuclear Reactions 119 B. Calculation of Radiation Widths The familiar formula for the rate of radiation of a quantum-mechanical system with dipole moment D radiating light of frequency v is Energy /second = -— | D»y | o c Introducing a self-evident notation for the type of multipole involved, and suppressing numerical factors near unity because of the roughness of our estimates, we may write the resulting width for electric dipole (2 x -pole) radiation: Bjit now we can estimate that, for a single particle moving in a region of radius R with charge e, the dipole moment is Z);/ = eR. For electric multipoles of successively higher order we can similarly estimate that, apart from numerical factors, the radiation intensity is reduced for each successive order in the ratio (R/X) 2 . Magnetic multipole moments are smaller than the electric moment of a given order by the factor (h/McR), which, using the nuclear radius value R = 1.5A^-10 -13 cm, leads to 21 x [l-3-5---(2Z + l)] 2 (94) 1 /RV ec r^ f I — 137 \X/ p mag ^ J , ] ji elec r^ Q Q2^ — Mp.eleo — \McR/ ~ These estimates will be reasonable ones for the low-lying states of nuclei if the independent-particle, Hartree-like model is not too mis- leading. (We have suppressed factors depending on I, which would arise in a more consistent calculation from the complex angular behavior of the higher multipole radiation.) It seems not inappropriate to apply these formulas for the rough computation of those long-lived low-lying states of nuclei which are responsible for isomerism. The detailed comparison with experiment is beyond the scope of the present work; we make no correction for the additional non-radiative transitions due to internal conversion, and so on. But it is useful simply to show that the radiation widths we compute would give the possibility of lifetimes like those observed for reasonable values of radiated angular momentum 120 A Survey of Nuclear Reactions [Pt. VI and energy. Table 9 shows lifetimes like those observed for the very transitions which the shell model predicts for the lowest states. TABLE 9 Lifetimes for Radiative Decay of Low-Lying States A = 100 A = 200 Multipole Energy = 200 kev Energy = 100 kev Electric 2 3 -pole . 029 sec 0.9 sec Magnetic 2 4 -pole 1 . 5 years 21 years We cannot expect the estimates of (94) to hold in the region where the level density is very high, and the idea of the excitation of a single particle no longer plausible. The sum rules themselves show that the estimate of a constant dipole moment, independent of energy, must fail ; the / sum would not even converge. A very rough idea of what to expect, consistent at least with the notion of a compound nucleus, may be gotten in this way. We expect the single-particle estimate to apply not to one single level of a highly excited nucleus, but to a whole group, dividing up the width, so to speak, among a great many levels each of which shares some part of the combining possibilities with the ground state. But, over how big an energy range must we spread the radiative width? A guess is afforded by the spacing of levels near the ground state, where it is not unreasonable to think that only a single particle has been excited. That spacing measures the energy region to be assigned to the excitation of one particle. Purely as a very rough orienting estimate, let us write for the radiative width of an average level in the region where levels are dense : 1 /R\ 21 r elec = _\ E x 137 \X/ 7 p(0) Ti™* = 0.02A~ 2A Ti elec P (E y )[l-3---(2l+l)] 2 (95) where p(0) and p(E y ) are of course the number of levels per unit energy (really of levels with certain fixed J values) at excitation energies and E y . The actual radiation from any level formed in a nuclear reaction will of course involve transitions not simply to a fixed state below, but to all combining levels at lower energy. The total radiation width then for a given type of multipole radiation can be written r t ot' ,e ' ec = Z Tf^iEi - Ey) all/ Sec. 9C] Radiative Processes in Nuclear Reactions 121 Replacing the sum by an integral, and using the form of (95), we get l,e P(0) P\E y max) E _2l+l P(E. y max x Ey) ClEy (R/hc) 21 137[l-3---(2Z+ l)] 2 (96) The integrand represents the gamma-ray spectrum immediately follow- ing decay, without taking into account any of the subsequent cascade gamma-rays. The most common example of such radiation is that following thermal neutron capture. The spectra as measured (K7) show very marked effects of transitions to a few specific low-lying levels, such as the ground state itself, and do not fit our statistical estimates very well. There is not much information about the wide gamma-ray spectra which do seem to underlie these special lines and which presum- ably correspond to what we have estimated (D2). We tabulate, for nuclei in various regions of mass number, the com- puted radiation width following neutron capture, for radiation of various multipole types. The level densities are taken from (31), and the whole calculation is very rough. There seems here, too, to be evidence against TABLE 10 Total Radiation Width after Neutron Capture: Various Multipoles A Max Ey = Bff (Mev) Widths in Electron Volts T\ elec p mag r 2 el J^mag 1 observed 60 120 200 8 7 6 4 0.4 0.1 5 X 10-3 0.3 X 10- 3 0.07 X 10-3 2 X lO- 3 0.2 X lO" 3 0.05 X 10" 6 30 X 10 " 7 2 X 10- 7 0.3 X 10" 7 0.2 the full contribution of electric dipole radiation. Probably there are such transitions; some appear to have been identified in light nuclei, but they do not exceed in probability either magnetic dipole or electric quadripole by as much as an order of magnitude. C. Photo-Induced Reactions From the theory so far given, the behavior of gamma-ray-induced nuclear reactions follows in a somewhat sketchy way. The general 122 A Survey of Nuclear Reactions [Pt. VI statistical theory describing the reaction as a two-step process is here applied : a y , p = S y (E)D p (97) where the cross section S y is the absorption cross section for the gamma- ray, and the factor D p , as in Eq. (40), describes the break-up, by emission of particle p, of the compound nucleus — here just an excited state of the target nucleus — exactly as in particle reactions. Below the binding en- ergy of a single neutron, no particle reaction can be observed, of course, since D p is exactly zero. Elastic and inelastic scattering of gamma-rays are possible, and would be governed by an analogue of the usual atomic dispersion theory, with the electric dipole moments perhaps not pre- dominant. As the particle threshold is crossed, D p rising from zero, the cross section <r yp rises as well, probably remaining rather small though increasing with a fairly high power of the energy, as a result of the im- portance of electric quadripole transitions. Only at energies where the excitation of the alpha sub-units becomes likely will the dipole moment assert itself; there the cross section S y rises to high values. As soon as a few Mev of excitation are available beyond the threshold for two- neutron, or even neutron-plus-proton, emission, this process will effec- tively eliminate the simple initial (7,^)- Thus a peak will be observed in the excitation function, with a width governed largely by competition. This peak, coming wholly from the factor D p , is superimposed on a fairly rapid rise and subsequent fall of the dipole cross section S y . There is still no clear division of the observed peak into the two factors; it is now fairly sure that the simple competition cannot be the entire reason for the width of a few Mev of the so-called "resonance" for the (y,n) reaction on a variety of targets (K4). The growth in neutron yield for a given gamma-energy, examined as a function of A, does, however, appear to be due mainly to the decreasing neutron binding energy and, hence, increased opportunity for two-neutron emission, as A grows. That the compound nuclear state formed by gamma-excitation might be of a rather special kind, with a particular internal motion, has been suggested (Cll, S10, T2). No clear evidence for this view has been produced which could not be duplicated on the present picture. The over-all { <r(E) dE, the presence of an apparent resonance due to com- petition, and the variation of neutron yield with A cannot distinguish the special motion from rather generalized features of any dipole ab- sorption. Good measurements of the peak widths and the relative yields of the various reactions are not yet at hand. That the assumption (97) is not entirely right seems demonstrated by the rather high yields of charged particles, especially of protons, from Sec. 10A] Nuclear Fission 123 gamma-ray-excited nuclei. These yields are far in excess of those calculated from statistical theory whenever, as for medium or heavy nuclei, the statistical emission of protons is much reduced by the Cou- lomb barrier. The protons tend to be distributed in angle more trans- versely than isotropically. All of this suggests that a process which contributes only a small part of the total reaction — for medium nuclei, the protons are only a few percent at most — can occur in which protons are as frequent as or more frequent than neutrons. Any process which amounts to the leakage of the protons out of a small region of the nucleus before that region has lost its excitation energy by conduction of the "heat" to the whole nuclear volume will do. A direct photoeffect, in which the single proton is ejected from the smooth nuclear potential well, is an extreme case of this view (B15). Something between this and the statistical equilibrium idea is more likely to be correct. Emission of alphas, deuterons, and other fragments seems to give further signs of the need for a more detailed theory. Reactions may be induced by the varying electromagnetic field of a charged particle. This problem has been discussed in terms of the method of virtual quanta, mentioned in Section 8, mainly for electrons. SECTION 10. NUCLEAR FISSION No nuclear reaction type has been so much discussed, and none has attracted so many workers, 1 as the curious reaction called fission. The fact that this reaction involved so profound a rearrangement of nuclear matter that neutrons were emitted in greater numbers than one per divided nucleus permitted the chain reaction, and thus the large-scale release of nuclear energy, even under terrestrial conditions of pressure and temperature. We shall discuss fission rather sketchily and semi- quantitatively here, with major attention not to detailed experimental results, but to the features of most general interest. All information here presented, without exception, is from the published literature, much of it from the spate of studies of fission in the first two or three years after its discovery (T6). This section does not pretend to be a full guide to the present state of knowledge of fission. A. The Energetics of Fission The fission reaction cannot conveniently be written as we have written all others: T(z,p)R. The projectile may indeed be a proton or neutron, or the reaction may be observed to occur spontaneously, but the product nucleus is not one of the light nuclear particles, with a heavy residual 1 See, for example, the semi-annual reports of the U. S. Atomic Energy Commission. 124 A Survey of Nuclear Reactions [Pt. VI nucleus. On the contrary, the products of fission are nuclei of sizable charge and mass, covering a range of scores of charge and mass numbers. In Fig. 18 we present the yield curve for several typical fission reactions (G3, Nl). The target nucleus has divided into two major fragments 7 U 235 + K ^ 6 — u 0) J \Bi 209 +d 1 lu 236 +w (3 S 4 JTh^l+a > "So * 3 a e & 2 O / 2 £ 1 7 1 ll \ 1 \ 50 100 Mass number, .A 150 Fig. 18. Percentage of occurrence of product nuclear fragment of mass number A, plotted against mass number, in three different fission reactions. The neutron fission goes with slow neutrons; the alpha-induced, with alphas of 38 Mev; the deuteron fission, with 200-Mev deuterons. Note the trend toward a symmetrical distribution, and the invariably large spread in fragment masses. (with a few light particles sometimes also in a kind of spray). It is clear that the description of the emission of a particle p from an essentially unaffected residual nucleus R is inappropriate. But the phenomenon can be pictured in a very simple and convincing way from the most fundamental ideas of the nuclear model, treated in a nearly classical fashion. 1 We have throughout referred to the nuclear matter as a kind of "quantum liquid" of nearly constant density, some 1 Bohr and Wheeler (B18) gave the first extended treatment of the theory; we follow them rather closely. Sec. 10A] Nuclear Fission 125 1.45 X 10 14 g/cm 3 . A heavy nucleus is then a small nearly spherical drop of this fluid, with radius R = ro-A^ cm. In such a nucleus, look- ing apart from the entire complexity of quantum effects, odd-even regularities, etc., we can recognize a static equilibrium for the con- figuration. The attractive forces which cause the drop to cohere are, just as in a drop of water, short-range forces for any given nucleon satu- rated by the interaction with its nearest neighbors among the nucleons. In the roughest way, then, the binding energy of a nucleus of mass A, charge Z is just proportional to the number of nucleons present, each forming its full number of bonds with other nucleons. The negative of the binding energy, — E B , we define as usual as the total energy of the neutral atom (nucleus plus its surrounding electrons) minus the total energy of its constituent neutrons and protons when removed to large separations, including the energy of the electron needed to neutralize each proton. Then stability means positive Eb. Then Eb ~ volume, and, assuming constant density, the volume energy alone gives —Eb = —avA. But, like any liquid drop, the nucleus has a surface. Even for the heaviest nuclei, many nucleons lie on the surface of the drop. Their bonds are not all saturated; there is a net deficiency in binding energy, a positive surface energy exactly like the surface energy of a water droplet, but clearly of the greatest importance. We then expect — E B ~ —a (volume) + b (surface), under the same constant density assumption. There is yet another classically evident contribution to the nuclear energy. This is the mutual Coulomb repulsion of all the protons in the nucleus. It is not a saturating force, with finite bond numbers and a short range. On the contrary, it is a long-range force to which all the protons contribute. On the most naive assumption of con- stant charge density, the Coulomb energy of a spherical drop of radius R = r A ^ is simply the integral /Ze\ /Ze\ rrdVdV _3Z 2 e 2 3 ZV Vol/ ' W/ JJ I r — r' 1 ~5~R~~ 5r A* ^ sphere Purely classically, then, the binding energy of a liquid drop of constant density and uniform charge density 1 is given by the form Z 2 -E B = -a v A + a s A % + a c — ^ (99) These simple ideas lead to a classical theory of fission. Suppose that now we have such a charged classical drop. If we slowly deform it by 1 For a careful discussion of this whole procedure, and of the limitations of these ideas, see (F2). 126 A Survey of Nuclear Reactions [Pt. VI elongating it, we clearly increase the surface area: this costs us surface energy. But the two halves of the drop are now found at a larger dis- tance from one another than in the spherical form. This means that their mutual Coulomb repulsion has decreased, and we have gained binding. As we deform the drop more, we increase surface energy and decrease Coulomb repulsion still more. Finally we can divide the drop into two remote spheres, each of half the original charge. In this state the Coulomb energy has been much reduced, and the surface increased. For a sufficiently high charge density, however, the Coulomb reduction must outweigh the surface energy increase, and the fissioned drops will be the stable configuration. The original sphere may then be in fact in a state of unstable equilibrium; any finite distortion from the sphere will cause the droplet to divide. Or it may be that, originally, small distortions into an ellipsoid will cost more surface energy than they gain in Coulomb energy, and the original sphere will be in a state of stable equilibrium. Yet finite displacements — distortion into something nearer a dumbbell shape — may so reduce the Coulomb energy that some intermediate distorted shape is a state of unstable equilibrium, capable of going either back to the original sphere, or on to two divided droplets. In this case the original drop is only relatively stable, and a genuine disturbance might cause fission. We shall see that this is the actual nuclear case at hand. 1. The Semi-Empirical Theory. We can make all these notions more precise. Let us go back to the drop-model energy content appropriate for the constant-density nucleus. We wish to evaluate the constants for volume, surface, and Coulomb energy. In addition, we shall add two terms which represent an effort to make the drop idea conform more closely to nuclear experience. It is fundamental in nuclei of course that the exact numbers of individual nucleons have a real role, unlike the molecules in any large-scale drop, where small fluctuations in mass have an unimportant effect. There are two reasons for the effect, and both of them reflect the essentially quantum character of the nucleus: first, the total number of particles is small, so that quite generally small changes in A could have important effect; and, second, the strong inter- actions seem to result in something like the atomic shells showing as fine detail on the general course of the liquid-drop energy content. We shall throughout disregard the details of this phenomenon. There is a second consequence of the special nuclear forces : the protons and the neutron seem to tend to group in pairs or even in alpha-particle-like units. There is a binding energy bonus for having the neutrons and protons equal in number. We shall introduce semi-empirically, then, two terms into our binding energy formula: Sec. 10A] Nuclear Fission 127 (1) A term, called the symmetry energy, which depends on the square of the difference between the number of protons Z and the number of neutrons N = A — Z. The fact that the term is quadratic represents the fit in first approximation to the trough of the valley in a binding energy surface, plotted against A and Z. The most stable isotopes lie in a crooked valley furrowing this surface; we imagine that the bottom of this valley can be approximated in cross section by a parabola. If we associate with each unpaired particle a definite energy, the whole term can be written (N - Zf a r (TV "t 6) A. (2) A term to stand for the last unpaired particle, expressing the fact that nuclei of even N and even Z are the most stable for a given A, and that those with N odd, Z odd are the least stable, while the odd-even or even-odd category is intermediate. This term may be empirically determined from the difference in energy between successive beta- transformations down a chain. With these additions, the semi-empirical formula becomes z * W - zf -E B = ~a r A + a s A V3 + a c ~ T .+ a r ± 8(A) (101a) A^ A We can evaluate ay and as by fitting the formula to the known mass- defect curve, which is by no means very accurate in the region above A = 40 or so. The procedure is made easier if we first evaluate a T in terms of a known a c . This we can do by fitting the valley of the stable isotopes, i.e., by finding the Z which corresponds to the most stable isobar for each A. The data fix this for odd Z at least to within about ±J^ for Z, making the valley pass between the two most abundant isobars. From our formula we need find only dM(A, Z)/dZ and set it equal to zero. This should mark the trough of the valley. [Note that M(A, Z) = -E B + AM n + Z(M H - M n ).] From dM/dZ = we get a relation between Z and A. The function so determined, called Z A , is given by the expression Z A _ (0.00081 + a T ) A ~ 2a T + 0.00125A % using ac = 0.584 Mev, as given below. If we smooth over the data, neglecting several kinks in the Z A function, we can find that, with ac as given, if a T = 0.083 mass unit, the fit to the empirical course of the curve is good. We consider throughout only nuclei above A ~ 20. A 128 A Survey of Nuclear Reactions [Pt. VI set of good values for all the constants then is „ 77.3 I A \ 2 -E B (Z,A) = -14.0A + 13.1A* + -7-\-z ~ z ) A A% lo;Aodd + 0.584 — ^ + \ ±6V A ' Z even, A even (ioib) where we have expressed Eb in Mev. A similar semi-empirical formula due to Fermi has been extensively tabulated (M7) for nuclei of all plausible Z and A, going far beyond the known range. The form there used does not display the Coulomb energy directly, but combines it with the symmetry energy, using as a reference the bottom of the actual valley of stability Z A rather than the artificial situation of equal neutron and proton number. Written for the atomic mass in mass units, the Fermi form is M(A, Z) = 1.01464A + 0.014A^ - 0.041905Z A + 0.041905(Z - Z A ) 2 /Z A ^ |±0.036M«; ! ° dd ' V Ven nm , + Z even, A even (101c) 0; A odd with Z A /A = 1/(1.9807 + 0.01496A % ). An earlier version of the formula is cited in a more accessible place (Sll). Our (101b) is in reasonably good agreement with the Fermi formula, but the latter (101c) is preferable for actual calculations. (See Section 3B in Part IV of Volume I.) A more elaborate treatment has been given which includes a measure of the compressibility of nuclear matter, i.e., a departure from the con- stant density assumption (F2) . The effect is quite small. The Coulomb constant ac is not in fact evaluated from the semi-empirical formula for a large number of nuclei, but is given from the comparison of the so- called "mirror nuclei," isobars with N — Z = ±1, where direct meas- urement of beta-decay energy has been made (Fl, F3). The radius it corresponds to is somewhat larger than that we have taken from high- energy nuclear reactions; it gives r = 1.48 and not 1.4 X 10~ 13 cm. This difference may reflect in part the tendency for the protons to move to the outside of the nucleus, or it may simply reflect the inadequacy of the rather crude semi-empirical formula. Let us now consider the energetics of fission with formula (101b). A very simple model of fission may be made by assuming that the drop divides into two drops, each of the same charge density, proportional to Z/A. Let one fragment have charge fZ, the other (1 — f)Z, and simi- Sec. 10A] Nuclear Fission 129 larly with the A' a. Then the energy released on such a fission, say AE(f), is given by the expression AE(f) = 13.1A^[1 -/« - (1 -/)«] Z 2 + 0.584 — [1 - f A - (1 - /)»*] in Mev (102) The volume and symmetry energies do not change; we neglect the small term 5 of (101), since we have not required integer values of Z or A. The treatment is entirely classical, and depends only on the assumed surface and Coulomb energies of drops of constant charge density. We plot in Fig. 19 the net energy release AE(f) as a function of / for nuclear drops of several charges. Note that, above a certain value of Z, fission will release energy for a wider and wider range of fragment sizes. For high enough Z, indeed, very small fragments — resembling alpha-par- ticles — may be released with a net gain in stability. This simple theory predicts the occurrence of fission and of alpha-radioactivity for high Z. Below the limiting Z value of course not fission, but combination of light nuclei, will represent a gain in stability. So far we have discussed only the energy difference between initial spherical drop and final spherical fragments. If this energy difference, AE(f), is positive, the process will proceed spontaneously. But the rate of fission, which will determine whether or not the process is ob- servable (rates corresponding to half-lives of more than Ty 2 ~ io 21-22 years are not now observable), is still entirely open. In Fig. 20 we have represented quite schematically this complicated process. We have plotted the energy difference between initial spherical drop and dis- torted drop against a single parameter — some measure of the effective distance of charge separation. Of course the distortion can in fact be represented only by many parameters. In our simplified plot, the value AE is measured from the initial energy to the final energy, from E t to E f . We have plotted four cases: reference to the figure will make them clear. In case (1) fission is not an exothermic process and cannot occur. In cases 2, 3, and 4, it is exothermic and occurs spontaneously. In all of these, the initial sphere is in equilibrium at sp under the mutual action of surface and Coulomb forces. But in case 4 the equilibrium is un- stable; any finite disturbance will cause fission. The actual nucleus would last only a time comparable with the characteristic nuclear times, for the zero-point oscillations would cause the needed displacement. Cases 2 and 3 show initially stable nuclei; small displacements will not lead to fission, but will be followed by a return to the initial spherical equilibrium. Only finite displacement, adequate to supply energy greater 130 A Survey of Nuclear Reactions [Pt. VI =M(Z, A) -M(Z,fA) 0.2 0.4 0.6 0.8 1.0 Fractional charge of fragment, / 1.2 Fig. 19a. Potential energy classically available for fission into two fragments of the original charge density, but charges fZ and (l-f)Z, plotted as a function of fragment size, /, for several nuclei. For heavy nuclei, division into fragments of a wide range of sizes is exothermic; for nuclei lighter than, say, bromine, fission is endothermic even with the most favorable fragment size, and symmetric with / = 0.5. Sec. 10A] Nuclear Fission 131 than the height of the barrier, at P, will lead to fission. Classically, it will never occur spontaneously; the initial state is a; state of relative stable equilibrium. Quantum-mechanically, in the actual nuclear case, spontaneous fission can occur, by the tunneling effect — diffraction beneath the barrier. Whether or not this will occur at an observable rate will depend on both the height and width of the barrier. In case 2, the barrier is high and wide; we may expect that spontaneous fission will not occur observably, and that large energies of excitation compar- able with AE would need to be added to such a drop before fission could occur. This might be the case of a nucleus well beyond the limiting Z near 35, but not one of the natural radioactive series. It is evident that the most useful application of these ideas would be the calculation of the barrier heights, or fission threshold energies. The potential energy contour of the drop plotted as a function of some repre- sentative parameters determining its distortion from the sphere (of course two parameters are all we can plot) is a surface with a flat portion far away from the center (where the two fragments are far apart), rising to a fairly high elevation as the frag- ments approach and feel the Cou- lomb repulsion; in the very center of the contour map, where the drop is almost spherical, there is a crater-like bowl in the center of which lies the original sphere before fission. Small displacements will never cross the lip of the crater, but there are passes in the crater wall, the lowest of which lies above the crater center by just the fission threshold energy. This pass, or saddle point, is what we want to find, in respect to both its height and its position on the map, i.e., to the shape of the drop at the unstable equi- librium position, where it can either go on to fission or return to the original sphere. (See Fig. 21.) , 20 40 60 80 100 Atomic number, Z Fig. 19b. Maximum potential energy classically available on fission into two drops of equal charge and mass, plotted as a function of atomic number Z. 132 A Survey of Nuclear Reactions [Pt. VI 50 100 Distance apart of nuclei centers (10" 13 cm) Fig. 20. Potential energy of nuclear drop as a function of distortion. The compli- cated distortion is here represented by a single parameter, some measure of effective distance of charge separation. The energy surface is plotted in the plane which represents the most favorable distortion, through the "pass" in the energy crater. Note the stable position at sp, the original spherical shape, the increase in potential energy to reach the pass at P, the energy threshold for fission, Eth, and the long decline down the curve to D and beyond. The distant part of the curves is simply the 1/R behavior of the Coulomb energy of spherical charges. If continued in until the fragments touched, any distortion being neglected, this would reach the points marked C, the end of the Coulomb barrier. How the drop distortion allows fission at much lower thresholds is seen from the bending over before point C. Sec. 10A] Nuclear Fission 133 134 A Survey of Nuclear Reactions [Pt. VI The full calculation is of course difficult. The number of degrees of freedom of a classical charged drop under axially symmetric distortions is very great, but some important consequences can be obtained quickly. (1) Let us consider the limiting case, where the charge density is small and the surface energy far outweighs the Coulomb repulsion. In this case, the drop will try to minimize its surface at all deformations, and the spherical shape will be stable for every distortion up to the very radical one of allowing the drop to be molded into two spheres and mov- Undistorted a; = 0.74 (a) (b) x = 0.65 x«l (c) (d) Fig. 22. A series of equilibrium shapes for the nuclear drop. Shape (a) is of course the original sphere; (b) the shape of the drop at the critical distortion for x = 0.74 = (Z 2 /A)/(Z 2 /A)n m ; (c) critical shape for x = 0.65; (d) the shape corre- sponding to a: « 1, division into two nearly tangent spheres, where the Coulomb tidal distortion is neglected (low Z). ing those spheres just out of the range of the surface forces. The shape of Fig. 22d is the case we shall compute. Here the Coulomb energy is not strictly zero, but only very small. The critical shape will then still allow for a tiny neck connecting the two fragment drops, which can be torn apart by the Coulomb repulsion. If we neglect the energy of the neck, which is very small in our case, we can simply compute the differ- ence between the energies of the two equal spherical fragments and the original sphere. The fission threshold energy, E t h, will in this case be E th = 2-4.Tr [©"*] M(A) 1A r ] 2 <r + 2 3 (Ze/2) 2 + 5(A/2)^ (Ze/2) 2 3 (Ze) 2 2(A/2y A r 5A A r (103) Sec. 10A] Nuclear Fission 135 (with <r = nuclear surface energy/cm 2 ) from which E th Z 2 e 2 /r = 0.260 - 0.065 -f- (104) MA H r ) 2 <r ' ' A iirr 2 a e 2 /r Z 2 _ 5 Coulomb energy where the parameter — • — is just - • . 4irr a A 3 Surface energy (2) Now we consider the other limiting case, with the original droplet so packed with charge that the slightest displacement from the spherical shape will produce fission. Here Eth = 0. To compute this we con- sider small constant-volume distortions, axially symmetric, without mo- tion of the center of mass, but otherwise arbitrary. We may represent the radius of the drop surface as a function of the co-latitude angle, 0, by a series in the orthonormal set of Legendre polynomials y(6) = 1 + 0-P x (cos 9) + a 2 P 2 (cos 0) + a 3 Ps(cos 0) -\ (105) R Here the coefficient of Pi (cos 8), oti, is set equal to zero, since for small displacements (where the higher harmonics may be expected to grow small), it corresponds to displacement of the center of mass. One con- dition is set on the motion by this requirement. The purely mathe- matical requirement that the function y(6) be single-valued for repre- sentation as a Legendre polynomial expansion does restrict the physical motion somewhat. Such a shape as that of the other limiting case — two near-spheres joined by a thin thread — cannot be so represented. This turns out to be an unimportant restriction; our answer is very far from such a shape for the observed fissioning nuclei. If we make the calculation of the classical area and Coulomb energy with assumption (105), and keep only the leading term, in a 2 , we obtain these results for surface and Coulomb energies of an infinitesimally dis- torted drop (81), neglecting all terms of higher order in the small co- efficients a n : 5A H r \5 / tfsurf - -E surf sphere = 4x4 Hr„ a , to . - (106) isurf - ^surf •" " = 4irA "ro- Then the fission threshold energy, measured in units of the undistorted surface energy, becomes + ■■■) (107) ±irA % l0 2 c 5 V # surf sphere 136 A Survey of Nuclear Reactions [Pt. VI We can read off the result: any displacement will lead to fission (i.e., Eth < 0) if JV phere /#surf Sphere > 2. Writing this result in terms of the semi-empirical constants, we have E c /E sm! | lim = Z 2 /A \ iim -a c /a s = 2 .-. Z 2 /A | lim = 45.0 (108) 0.30r- 0.20 — 0.10 o.e l.o x =E c /2E a =CZ 2 /A)/(zVA) lim Fig. 23. The fission threshold energy, Eth, as a function of the parameter x. The dimensionless function /(x) is just Eth measured in units of the nuclear surface energy in its undistorted spherical form, and x is defined by the relations x = Ec°/2E S ° = Z 2 /A/(Z 2 /A)um- From fitting the experimental data for U 239 , we get (Z 2 /A) v , ■ 48 (see text). The function f*(x) is the energy difference between the sphere and the shape of Fig. 22d, two equal nearly tangent fragments, again meas- ured in units of the surface energy. For small x, below the vertical stroke on the curve of f(x), the function f(x) is taken from its limiting value for small Z; above the right-hand vertical mark, f(x) has been calculated numerically. Between the two strokes f(x) has been simply interpolated free-hand. We can of course continue the series still further, and examine how Eth/E aur f spheie , which we will write as f(x), behaves as a function of x = Ec/2E S . This has been done out to quite high powers of the small Sec. 10A] Nuclear Fission 137 coefficients — with as many as six or eight harmonic terms. A numerical calculation, done with high-speed calculators (Fll), has been carried out with ten harmonic terms, and without making any power-series expansion in the coefficients. The function fix) is known near x = from the calculation in case 1 above, and near the limit x = 1 we have plotted the calculated results in Fig. 23. The dashed part of the curve is a reasonable interpolation. We can regard x either from a theoretical point of view, as the ratio Ec/2E S , or in a form more directly comparable with experiment, as (Z 2 /A)/(Z 2 /A) ]inl . In the latter form, we see that we can read off the value of the fission threshold E t h for any nucleus by computing the surface energy asA 2/3 , noting that E t h = f(x) ■ as A y *. We can now apply these results best not by taking the limiting {Z 2 /A\ Ta entirely from our semi-empirical constants, but by fitting one observed fission threshold (S8) to the fix) curve. If we fit E t h (U 239 ) = 6.67, and take, for the surface energy for U 239 , 502 Mev, we obtain f[x(U 239 )] = 0.0132. This yields Z 2 /A\ Um = 47.8, which can be com- pared to 45.0, obtained by taking the constants directly [Eq. (108)]. 2. Quantum Effects. We have treated the problem wholly classically. It should be observed that this neglects two quantum-mechanical effects. First, fission may take place for excitation energies below the fission threshold by reason of the tunneling effect; if the representative point comes near the pass over the crater lip, it will often go through the thin barrier remaining. In addition, the vibration of the drop in the dis- torted mode will have a zero-point energy. Both of these effects can be estimated by computing how the barrier looks as the representative point moves in its path over the energy surface. This can be done under the assumption that the drop distorts through the sequence of shapes near the saddle-points for larger values of x. With this assumption, Frenkel and Metropolis (Fll) calculated the barrier as a function of distortion. The Gamow penetration could then be estimated by using this potential barrier, and an effective mass estimated from hydro- dynamic arguments. They obtained a penetration probability P for various E th given by P = io -7 - 85 ^. The zero-point energy can be calculated from the shape of the crater near zero distortion. This gives a result of some 0.4 Mev for the uranium case and leads to an estimate of the oscillation frequency. Such a zero-point energy, small compared to the excitation involved, implies that the motion can be represented classically as a rather well-defined trajectory by building up wave packets from the actual quantum states. The threshold estimate is then made as follows : -"th (Classical) = -£/ z ero-point l -* neutron ~T~ -'-'neutron binding 138 A Survey of Nuclear Reactions [Pt. VI Using the results that neutron binding in U 239 is 5.2 Mev (from the semi-empirical formulas) and that the measured neutron kinetic energy is 1.0 Mev, we get the 6.6 Mev used above for E t h (classical). Only qualitative results can be expected from this theory. The actual fission is not into symmetric fragments at these excitations, and the fission thresholds do not in detail vary from nucleus to nucleus in the smooth way here predicted. Moreover, the spontaneous fission rates are in poor agreement. It must be observed that the very large exponents which occur mean that a small error in the barrier will lead to enormous errors in the rate calculations. Perhaps only the logarithm of the decay rate should be expected to be of some significance, within a small factor of the truth. Arguing the other way is more sure. It turns out from the f(x) curve (it will be an interesting exercise for the reader) that for Z > 98 the typical isotope would be radioactive to such an extent that it would probably disappear from the earth in geologic time, not because of the accidents of alpha-particle disintegration rates (which vary both up and down in just a few Z or A units), but because of the inescapable effects of fission. Thus the fact that the heaviest nucleus occurring in nature is uranium is a consequence only of the kinks in the binding energy curve, coming from the closed shells at Z — 82, N = 126; this means that the next few elements have short lives against alpha-decay. But the more fundamental reason that no ele- ments of Z = 100 or up are found is certainly fission. It will be inter- esting to see the first nuclear species whose lifetime is controlled by spontaneous fission. There is one more deduction from this simple theory which is of inter- est. The calculation of Eq. (104), which we can plot as the function f*(x) for < x < 1 (see Fig. 23), gives the energy difference between the original sphere and the nuclear matter arranged as two equal frag- ment spheres just touching. Now the energy released on symmetrical fission is a given function of Z and A, independent, of course, of the path followed in dividing. If the released energy is Ai?(3^), then the energy difference between the critical distorted shape and the separated fragments is just f(x)asA ^, while that between separated fragments and two tangent fragment spheres is f' t (x)asA 2A , which is, from Fig. 23, some 25 or 35 Mev greater in the region of the fissionable nuclei. This means that, when the distorted surface finally tears to release the two frag- ments, which when far away will eventually come to their stable spherical shape, the difference of about 30 Mev is stored as energy of distortion — energy of excitation — in the two fragments. It is, moreover, plain that the reverse of fission — union — will in general require much more kinetic Sec. 10B] Nuclear Fission 139 energy than that released in fission. It will be very unlikely that the special distortion which leads over the low pass in the crater lip will be achieved in the random collisions of nuclei. B. The Products of Fission The most obvious consequences of the simple theory given above are in fact not observed. The fission fragments do not appear as equal nuclei with / = }/%. On the contrary, the distributions of Fig. 18 are those observed experimentally. No clear explanation of this fact has yet been given. It seems almost sure that there is no complex distortion type which leads to a low-lying pass for some asymmetric distortion of the classical drop. It is much more probable that again we have to do with a reflection of the nuclear shell structure, permitting some motions, some ratios of charge to mass in division, more easily than others. The closed-shell N's at 50 and 82 do seem to mark near-maximum yields. There is some evidence that isotopes of those stable elements with highest N — Z are favored, but the expected predominance of even products is not yet found in the admittedly complex chains of fission fragments. It should be obvious by now that the initial products of fission are not in general stable nuclei, but are very neutron-rich, and begin chains of beta-decays to reach the appropriate ratio of Z/A for their smaller A (B18, W4). It has been shown that, in this beta-decay chain, excitations are in some cases in excess of the neutron binding energies for the product nucleus of a given beta-decay. This leads to the emission of delayed neutrons whose time of emission after fission is determined by the preceding beta-decay processes. Much more numer- ous are the so-called prompt neutrons, which emerge after a time short compared to any possible beta-decay. It seems very likely that these neutrons are evaporated from the highly excited fragments as they fly apart after the fission act. This would imply that some two or three neutrons are emitted, and that their energy spectrum and angular dis- tribution would be the quasi-Maxwellian one of our statistical theory (compare Fig. 6). The fission fragments, however, continue in motion at their high speeds of near 1 Mev/nucleon for a distance of a few milli- grams per square centimeter, or some 10~ 3 cm, in solid matter. The evaporation will occur with neutron widths measured in many kev at least, or in times short compared to the fragment time of flight. So the neutrons will evaporate isotropically from the moving fragment, and in the laboratory system will be emitted preferentially forward, since their kinetic energy will be quite comparable to that of the frag- ment, expressed as Mev per nucleon. Some evidence has been obtained to confirm this picture as well (W9). 140 A Survey of Nuclear Reactions [Pt. VI The products of fission, then, are mainly the two large fragments and the prompt neutrons. The fragment decay leads to beta-particle chains, gamma-rays, and the delayed neutrons. But there are other charged particles which emerge at least occasionally upon fission. In about one fission in five hundred an alpha-particle forms the third fragment, with kinetic energy from a few to about 25 Mev (M2). It seems well estab- lished that these smaller fragments actually originate from the com- pound nucleus, and from a tendency to come off at right angles to the direction of the main fragments, and that they arise during the act of fission. The systematic change in shape of the fission fragment size-distribu- tion curve with energy of the bombarding particle is the most striking feature of Fig. 18. Apparently, as the excitation exceeds the fission threshold more and more, the tendency toward the symmetrical fission expected in the simple theory becomes stronger. Slow-neutron uranium fission is never symmetrical ; at energies around 100 Mev, the symmetri- cal division is the most probable. Direct measurement of the ionization energies of the fragments confirms the observation on the masses of the fragments (J4). Whatever special effects produce the asymmetry are increasingly unimportant as more energy becomes available to the vibrating drop. It does not seem true, however, that the fragments of an asymmetric division have total fission product energies higher than the fragments of nearly symmetric division; there is even some tendency toward the reverse. C. Fission Cross Sections Fission, as a mode of decay of a compound nucleus, should be produced by any means that excited the nucleus above the required threshold energy. This has been verified for slow and fast neutrons, for protons, deuterons, and for alpha- and gamma-rays. In general the cross section can be computed by our usual methods — simply writing the cross section for formation of the excited compound nucleus by a particle of energy E v , and following it by the factor for its competitive disintegration: i{W)YAW) *,(E P ) = ,(E P ) " ' " > (109) */ T An "T J- 7 T ' ' ' where 1/ is the width for fission from a compound nucleus of excitation W. The calculation of the fission width is the calculation of how the excitation of the nuclear drop is distributed among all its possible modes of motion: there is available a large amount of phase space for motions which lead to distortions from which the drop cannot divide; only a small volume in phase is assigned to such motions of the point repre- Sec. 11] Nuclear Reactions at High Energy 141 senting all the parameters of distortion as do lead over the pass in the wall of the energy crater. Below the threshold for fission, the excitation energy can be lost by any other energetically available means, usually gamma-radiation or neutron evaporation, but not by fission. As the excitation W = E p + Eb(p) increases to near the classical fission thresh- old, tunnel effect can lead to fission. The fission width will increase, and fission will become a competitive reaction. We can expect the fission width to rise rapidly, perhaps something like exponentially, as the fission threshold is considerably exceeded. What the cross sections will be then will depend largely on how high the fission threshold is relative to the excitation energy of a given mode of excitation. Thus <rf(E n ) for U 238 + n 1 is zero until a threshold of about 1 Mev, from which it rises slowly to a value comparable with irR 2 . It stays nearly constant because for some time 1/ and r„ rise about in the same way. We know from the experiments with very fast fission, on elements like bismuth, where the fission threshold is some 12 Mev or more, that the neutron specific width rises more rapidly than that for fission. Very high excitation in Bi 209 leads mainly to multiple neutron evaporation; some ten or twelve neutrons are boiled off before fission becomes a very important competitor. Only when the fission threshold has come down, because of the gradually increasing value of Z 2 /A in this neutron boiling- off, does a-/ begin to grow to a value comparable with unity, actually about 0.1 or 0.2. This is confirmed both by the fission-product dis- tribution (which is that of fission by a nucleus of A ~ 200) and by the fact that the fission fragments come apart with energy near that ex- pected from fission near the threshold energy. The distribution of fragment sizes becomes broader with increasing excitation; the mode of division does not need to be so special. Even so, it is plausible that there are many more ways for a highly excited drop to emit neutrons than to divide by fission, so that, while the I> rises rapidly, it does not rise so rapidly as the other terms of total r, almost all of which, at these ener- gies, arise from neutron evaporation. Special processes, like photofission and fission in the region of applica- tion of the dispersion formulas, have been discussed in the literature (B18, G3, Nl). SECTION 11. NUCLEAR REACTIONS AT HIGH ENERGY The processes we have discussed so far are characterized by the fact that the energy of the incoming nucleon, both potential and kinetic, becomes quickly shared among all the particles of the nucleus. After a relatively very long time, during which numerous nucleon collisions 142 A Survey of Nuclear Reactions [Pt. VI occur, this energy may again become concentrated in a particular nucleon, or in some special mode of motion, and the nucleus will de- excite by particle emission, fission, or the slow processes of radiation. As the energy brought in grows, the number of available modes of disintegration naturally increases, and the lifetime of the compound nucleus becomes shorter and shorter. With energies in the range up to some 30 to 50 Mev, the nucleus will emit typically several nucleons before it cools, and the reactions are increasingly complex. Such reac- tions as (p,3n) and (p,4n) have been studied (K5). But, as the energy increases, there is a gradual transition to a state of affairs which is best described in a conceptual framework quite the opposite from that suited to the compound nucleus. This high-energy point of view works best above some hundreds of Mev of incoming nucleon energy, and the two schemes will merge in the broad transition region between. Most characteristic of the high-energy region is the fact that the mean free path, in nuclear matter, of the fast nucleon is comparable with the radius of the nucleus itself. The nuclear matter is no longer a black, impenetrable obstacle, which traps the incoming particle, but a kind of sphere of dilute gas, through which the fast-moving nucleon has an appreciable chance to penetrate without any collision whatever. Even a collision or two do not by any means rob the incoming nucleon of most of its energy. The momentum transfer in such a collision, which is typically transverse to the classically pretty well-defined trajectory of the short-wavelength incident nucleon, is of the order Ap ~ h/R, where R is the range of nuclear forces (S7). This implies an energy transfer of some 30 Mev. In the short time of such a collision (~10 -22 sec) the struck nucleon will make only very few, if any, collisions with the neighboring nucleons of the nucleus, and, although the momentum trans- fer is not so large that we are wholly safe in regarding the struck nucleons as free, still that will be a reasonable initial approximation. We shall see that the principal effects of the nucleon environment are conse- quences of the Fermi degeneracy of the nucleon matter. The collision will be modified by the impossibility of leaving the struck nucleon in a momentum state already occupied by some other nucleon of the nucleus. The geometry of the approach of the nucleon will determine the se- quence of events. If it passes through the nuclear edge, it may make no collision, or perhaps one. A very fast nucleon, having lost only 30 Mev, will emerge. The nucleus left behind may be excited by the full amount of the energy given to the struck nucleon, or by very much less if the initial path of the struck nucleon allowed it as well to leave the nearby nuclear surface without further collisions. A fast particle striking Jhe very center of the nucleus, where the nucleus is thickest, may have Sec. 11 A] Nuclear Reactions at High Energy 143 to travel a few mean paths in nuclear matter, and will typically make several collisions, leaving in its wake a few nucleons of 20 to 30 Mev energy each, perhaps losing all its energy, being finally captured. The secondary nucleons in turn can escape only if they are close to the nuclear surface and moving outwards. Some may do so; others will collide with many nuclear particles, and gradually spread their energy around among many modes of motion, forming a heated compound nucleus, which will cool off just as in the reactions of lower energy. A typical many-particle nuclear reaction, often called, from its appearance in nuclear emulsion photographs or in the cloud chamber, a "star," can then be thought of as a kind of approach to thermal equilibrium, be- ginning with a well-defined single-particle trajectory, followed through a cascade of secondary, tertiary, etc., collisions, and ultimately resulting in a diffusion of excitation energy like heat through a conducting sphere. The initial stages form the high-energy limit; here the reaction is to be described by a step-by-step following out of the nuclear cascade, taking into account collisions with one or with a few correlated nucleons; finally, the energy is shared by the collective motion, in something like transient thermal equilibrium, which is the low-energy, compound nucleus picture. A single star event may exhibit both features; it is the fact that the first stage takes a time ~10 -22 sec, very much less than the 10~ 16 sec of the last stage, which permits a more or less sharp division into the two stages. On this sharp division, which is only approximate, the simplicity of the picture depends. The intermediate time, during which the energy is distributed among too many degrees of freedom to be described in detail, but too few to approach any sort of quasi- equilibrium, is beyond the present theory; such complex cases will be especially important in the lighter nuclei, and for a mass A nucleus at energies 10A Mev, enough to dissociate the whole nucleus. Our treat- ment will deal wholly with the two well-separated stages: the initial cascade and the final thermal equilibrium. A. The Nuclear Cascade The initial cascade has been followed out by the most direct means. The theory has been the semi-empirical one of the so-called Monte Carlo method (G5), in which after the establishment of the model and the appropriate cross sections— which are the decisive features for the accuracy of the results— the calculation is done by considering the successive events in the motion of the incoming nucleon and all its collision partners, with their collisions in turn; choosing the actual steps by a series of random choices; and finally cutting off the whole process when some arbitrary low-energy limit is reached beyond which the energy 144 A Survey of Nuclear Reactions [Pt. VI is assigned as equilibrium excitation of the compound nucleus. This step-by-step process is extremely tedious, and only a few hundred indi- vidual "stars" have been followed through on paper. The results are subject then to a fair degree of statistical uncertainty over and beyond shortcomings of the model. But enough has been done to make the general features of the high-energy stage in star formation fairly clear. 1. Fermi Gas Model of the Initial Nucleus. The short collision time and high recoil energy make most plausible the use of the simplest of V J Fig. 24. Diagram of the potential well in which the nucleons are bound in the sim- ple Fermi-gas model. The Coulomb barrier affects only the protons. The well depth is determined by the nuclear density and the assigned binding energy. nuclear models, that of the Fermi gas of non-interacting nucleons con- tained in a well of assigned diameter and depth. Naturally the correla- tions actually arising from the nuclear forces, here ignored, will be important eventually. One clear sign of this is found in the process of the pick-up of a nucleon from the nucleus, to emerge bound to another outgoing nucleon in an emitted deuteron, a process far more important than the Fermi gas model would predict. But, aside from such details, it is appropriate to begin with a Fermi gas. The procedure is the familiar statistical one for finding the distribution of a completely degenerate Fermi gas, at T = 0°. The nucleus is re- placed by a spherical potential well in which the particles move without interacting. Each particle has a wave function which is simply a plane wave; the finite depth of the well, and hence the leaking of the plane waves into the classically forbidden region outside the well, is neglected. Then a particle of each spin is placed in every momentum state until all particles are accounted for. The neutrons and the protons are re- garded as entirely independent and as moving in separate wells; the additional Coulomb barrier for the protons may be regarded as beginning at the well edge. Figure 24 shows the model schematically. Replacing Sec. 11 A] Nuclear Reactions at High Energy 145 the sum over the filled states by an integral, and remembering that (2s + 1) particles with different spin orientations may be placed in each cell in phase space, of volume (2xh) 3 , we obtain for the number of particles of a given kind, with spin s, in a sphere of volume V, N s (2s + l)p 2 dp cK2/(2xh) 3 where p F is the limiting momentum to which the states are occupied, and the energy corresponding to pf, the Fermi energy E F , is given by E F = p F 2 /2M a . If we take the sphere of nuclear matter to be a sphere of constant density independent of the mass number A, as is usual, and write for the radius of the nucleus R = r -A H , we can easily find an expression for the Fermi energy for the nucleons of one kind: E Ft = (18tt) k /8 X h 2 /Mr 2 ■ (N s /A) H , where N s is either the atomic number Z for the protons, or the neutron number N = A — Z for the neutrons. This gives the numerical result: E Ft = 9.7(N s /A) 2A (a /r ) 2 Mev, where we have written a /r , the ratio of a , the classical electron radius, 2.82 X 10~ 13 cm, to the nuclear radius parameter. For orientation a few values are listed in Table 11. The well depth is now fixed to give the TABLE 11 Proton and Neutron Fermi Energies for Various Nuclei Ef (protons) Ei?(neutrons) Cu 65 n> 1.4 22.6 26 1.6 17.5 20 Cs 1: 1.4 22 27 1.6 16.9 20.7 1.4 21 28 1.6 X 10" 1S 16.1 Mev 21.5 highest-lying neutron state about 8 Mev binding energy, and an addi- tional Coulomb barrier may be assigned for the protons. This implies a nuclear potential well for neutrons in lead about 35 Mev deep, using the smaller value of the radius parameter, and an additional Coulomb barrier V c for the protons about 0.5(Z/A M )(a Ao) Mev high, or <~14 Mev for lead. In the calculations the two types of nucleons are often represented by a single gas of about average properties. In the region of high energies, the simple Fermi gas model has two characteristic consequences which are certainly to be expected even from more realistic models of the internal nucleon motion. It is perfectly clear that the complete neglect of nucleon interactions is an extreme procedure which tends to underestimate the range of momenta repre- 146 A Survey of Nuclear Reactions [Pt. VI sented in the statistical motion of the nucleons. One can regard this lack of a high-momentum tail as implying either the neglect of spatial correlations, in which the nuclear density fluctuates toward higher values because of the nucleon interaction and the transient formation of sub-units of two, three, or four nucleons, or as a neglect of occasional transitions, made possible by nucleon interaction, from the lowest states of the Fermi gas to some of the higher unoccupied momentum states. Both ways amount to the same thing; they form the basis for regarding the Fermi gas calculation as a first step in a systematic approxi- mation method. We shall look here, however, into the consequences of the raw Fermi model as they affect high-energy interaction events. (a) Influence of the Exclusion Principle. In this model, the incoming nucleon, once it enters the nuclear volume, fills a hitherto unoccupied momentum state. Now it enters into a collision with some one of the nucleons already present, lying in one of the filled levels below the limiting Fermi momentum p F . The spirit of the model, representing all nucleons as free, now clearly requires that the momenta of the two collision partners after the collision lie in an unoccupied region of the momentum space. A simple exchange of momenta corresponds to an elastic collision. Any other event must begin by fulfilling the condition of entering fresh momentum space regions. Evidently a small momen- tum transfer to the struck nucleon is forbidden unless the nucleon ini- tially lies very near the edge of the sphere in momentum space of radius p F . As long as momentum transfers not very great compared to p F are of importance in the process, which means that the incident particle may have momentum up to many hundreds of Mev/c (or even con- siderably higher if forward scattering alone is under consideration), the cross section for collisions with the average nucleon of the Fermi gas will be much reduced by the requirements of the exclusion principle. This will evidently increase the chance of passing through nuclear matter without collisions, and will tend to reduce the relative probability of small-angle scattering, both near zero degrees in the center-of-mass frame and, because the projectile also satisfies the exclusion principle, near 180° as well. Figure 25 presents the geometrical considerations in momentum space. A calculation (G5) on this purely geometrical basis shows that the exclusion principle reduces an assumed isotropic cross section for scatter- ing in the ratio: , - 2 \ _ / 7 PF 2 \ 0"av — Cf r ee II—- — - 1 \ 5 pi 2 / where o- free is the isotropic scattering cross section per unit solid angle for a free nucleon, and p t is the incident nucleon momentum. This formula Sec. 11 A] Nuclear Reactions at High Energy 147 holds only for values of p t > 2 V2 p F ; below that incident momentum the result is more complicated. The direct analytical evaluation has not yet been performed for a case with the cross section anisotropic. The angular distribution can be approached through the same geometrical scheme, if the scattering cross section is taken as a function of the mo- mentum transfer | p { - p f | = | 5 | only, which corresponds to the Born approximation without exchange. Fig. 25. Kinematics of the collision between an incoming nucleon and the nucleons of the Fermi gas. The incident, recoil, and final momentum vectors are shown, with spheres of radius in momentum space just p F , the Fermi momentum. The shaded region is excluded; no collision in which the vectors Precou or p flna i terminate within this region is permitted by the Pauli principle. (b) Influence of the Zero-Point Motion. The angular distribution of particles scattering from the Fermi gas involves two distinct and im- portant effects: the exclusion of low-momentum transfer collisions, and the smearing out of the otherwise unique correlation between scattering angle and energy of emerging particle, which can be regarded as the effect of the zero-point motion of the struck nucleons of the Fermi gas, which of course spread over a wide range of momenta, and hence of relative velocities, as seen from an incoming nucleon of fixed energy. The geometry of the event is just that of Fig. 25, and one can write rather complicated expressions for the cross section in the laboratory system for scattering a particle into the element of final momentum of the scattered particle, p/ - . ¥- = 'WtPF, P/, 6) (HO) 148 A Survey of Nuclear Reactions [Pt. VI where/ is a function of the momenta and scattering angle, and the regions of integration for the total cross sections can be found from the figure. Since the cross section for neutron-proton scattering, for example, is far from simple in the relevant energy regions, only numerical calcula- tions have been made. Figure 26 shows the general results to be ex- pected: the curves correspond to special choices for the cross section. - 10°/ \ - 45° 20V'' / V / \ \ \^ 50 Energy (Mev) 100 Fig. 26. The energy distribution of nucleons scattered at a fixed laboratory angle by a single collision with a nucleon in the nucleus. The effects of the Fermi motion and of the exclusion principle have been included. The incident energy is 100 Mev; an empirical fit to the neutron-proton scattering cross section was used. The vertical strokes mark the energy expected at the three angles shown if the collision was with a free stationary nucleon, in which case the energy would be a unique function of angle. The spreading effect of the Fermi motion is conspicuous (65). The smearing of the energy-angle correlation and the shifting of the scattering toward right-angle scattering is at least made qualitatively evident. All this applies of course to an event in which only a single primary collision is made within the nucleus. 2. Following the Cascade: The Monte Carlo Method. The absence of any theoretically based analytical form for the nucleon-nucleon scattering cross sections, and the rather complicated geometry in coor- dinate and momentum space required to treat the problem, have led to the use of an interesting method which enables not only calculation of the results of a single collision, but also actually follows out step by step the development of what may be called a nuclear cascade. In this application of a general computational scheme, called for obvious reasons the Monte Carlo method, a single incident nucleon of given energy is allowed to enter the nuclear volume. Its collisions are Sec. 11A] Nuclear Reactions at High Energy 149 followed in complete kinematical detail, and the subsequent collisions of all its collision partners are followed as well until either: (1) any of the excited nucleons passes outside of the nuclear volume, after which of course no further interest attaches to it except that it is an emitted nucleon, with its energy, direction, and charge; or (2) any of the nucleons involved, while still within the nuclear sphere, reaches an energy below some previously selected value, after which it is regarded as captured, and its excitation energy passes eventually into the motion of an excited compound nucleus. Then follows the process of nucleon evaporation, regarded as separate from the initial cascade. The cascade computations are usually made graphically. The distinguishing feature of the method is that the decision as to which of a set of equally probable events occurs is made by some procedure of random choice, using tables of random numbers, spinners, etc. Various simplifications of geometry may be made, and the continuous functions of angle, etc., which occur are typically stepwise approximated by dividing the interval into a set of discrete subintervals, giving the function some constant value as an approximation throughout the whole of the subinterval. Let us follow a neutron of energy E incident on a nucleus of fixed A and radius. Where does it make its first collision? Its point of entry is chosen by making a random choice among the rings of equal area normal to the direction of the beam. Then its distance of travel is found by assuming that it moves with a mean free path given by the assumed total interaction cross section with free nucleons of the given density (the effect of the exclusion principle can come in only after the collision has been made). A distance 5 which is small compared to X, the mean free path in nuclear matter, and which gives a probability of collision 5/X, can be chosen, and a random decision made as to whether or not it collides in this line segment. The nucleon continues to progress segment by segment in a straight line, until either it leaves the nuclear volume or it makes a collision. Typically at these energies a fair fraction of the nucleons traverse the nucleus without collision, especially those which enter near the edge. When the nucleon has made a collision, a whole set of random choices must be made. First, the momentum of the struck nucleon drawn from the Fermi gas must be found, by pick- ing at random from intervals of equal probability in momentum space, weighting the flat Fermi momentum distribution by the dependence of the collision cross section on relative momentum. Then the angle of scattering must be found. Usually it is best to work in the center-of- mass system, and to approximate the differential cross section for scatter- ing by a dozen or so steps. Once the angle of the collision has been found by a random choice, the final momentum vector of each of the collision 150 A Survey of Nuclear Reactions [Pt. VI partners is fixed uniquely by the conservation law, since the collision is regarded as free. Now one may see if the collision is allowed by the exclusion principle. If both the final vector momenta do not lie outside the filled sphere, the collision is nullified, and the original particle con- tinues undisturbed until it makes another collision. If the collision is permitted, then each of the collision partners is now a nucleon of definite vector momentum at a specified point in the nuclear sphere. The history of each can now be followed until it in turn escapes or is absorbed. Account is of course kept of all partners in each collision, and the event associated with one incident nucleon is not complete until every struck nucleon has been emitted or absorbed. It is obvious that the com- putation is tedious; it is equally clear that it is an extremely flexible method, capable of high accuracy. It is hardly a calculation so much as an experiment in thought, using a well-defined model. Two calculations have been worked out on these lines for incident energies 90 and 400 Mev, and reported in some detail (BIO, G5). They are unfortunately not strictly comparable, mainly because different fundamental cross sections for collision were used. In one case the target nucleus was lead, whereas in the other it was A = 100, more like the heavy nuclei of photographic emulsion. In Fig. 27 is drawn a typical cascade event just as it was followed. Here the approximation was made of treating the nucleus as a two-dimensional circle, after the first collision; the graph then gives directly the projected angles of emission. In the event shown a 400-Mev nucleon entered, making its first collision about one-third of the way across the nucleus. The two partners of this colli- sion were nucleons initially of 431 and 13 Mev of kinetic energy. It will be noted that a nuclear potential well of 31 Mev is assumed, with an additional barrier of 4 Mev (half the Coulomb barrier, since protons and neutrons are not distinguished). One of the first pair of collision partners leaves the nucleus directly, as a fast cascade fragment of 282 Mev. The other goes on to try to make a collision in a short distance, at the point marked by a circle, but it is forbidden to do so by the exclusion principle. Then it goes on to an allowed collision with a rather fast nucleon of the Fermi gas. In all, this event consisted of six nucleon collisions: one nucleon enters the nucleus with 400 Mev kinetic energy outside; and three leave, taking with them over 300 Mev, mostly in the kinetic energy of the one very fast collision partner of the first collision ; while four more are excited but "captured," leaving the residual nucleus excited by 66 Mev, with the mean binding energy per particle taken at about 8 Mev. The collection of results for both calculations gives some impression of the nature of nucleon cascades. The mean excitation energy of the Sec. 11A] Nuclear Reactions at High Energy 151 nucleus changes very slowly with the energy of the incident particle: the 90-Mev neutron on lead left a mean thermal excitation energy of some 44 Mev, the 400-Mev proton on emulsion nuclei only about 50 Mev. This slow rise of excitation with incident energy is partly an artifact of the somewhat forced comparison, but it is actually to be 28 Mev 7 Mev 400 Mev 282 Mev Fig. 27. A projected diagram of a nuclear cascade event, followed by the Monte Carlo method. The entering nucleon has 400 Mev of kinetic energy; the numbers mark the energy of each of the nucleons participating. The path of each nucleon is shown until it leaves the nucleus, or until it reaches an energy below 35 Mev, taken as the threshold for capture. Along the path of each particle an effective collision is marked with an open circle; a cross implies a collision which was for- bidden by the exclusion principle. Note that in this event three cascade particles emerged, all in the forward hemisphere, and the nucleus was left excited thermally with about 65 Mev, enough to evaporate several neutrons isotropically before be- coming stable. From (B10). expected as a general result, since the high-energy nucleons make high- energy secondaries, which typically depart from the nucleus with most of their energy, since the mean free path rises with nucleon energy. In both calculations also there is a rather wide distribution of energies among the cascade products, and a considerable fraction of the energy almost always goes to them, as the low mean excitation indicated. With the lower energy and the bigger nucleus, however, about 5 percent of those incident particles which made any collisions whatever left their whole energy in nuclear excitation, with no cascade particles emitted. In the high-energy, A = 100, case, no event of this kind occurred at all 152 A Survey of Nuclear Reactions [Pt. VI (among sixty studied); the minimum energy found in the cascade particles was about half the incident energy, 200 Mev. The general nature of the cascades can be seen from Table 12 and TABLE 12 Nuclear Cascades Studied by the Monte Carlo Method 90-Mev 400-Mev Neutron Proton on Pb on A = 100 Fraction of incident particles traversing nucleus without any collision 0.15 . 33 Fraction of cascades without emerging particles 0.05 Mean thermal excitation energy 43 Mev 50 Mev Mean number of cascade particles emerging per cascade event 1.2 3.2 Maximum number of emerging cascade particles 3 <~6-8 Figs. 26 and 27. It should be remembered that the sampling error alone in all these calculations is still quite large. Only a few hundred cascade particles have been "seen to emerge," and information about any special category of them is necessarily only qualitative in character. It is quite clear from the results that this process does not much resemble the evaporation model. The large number of emitted particles with quite high energy, and the strong departure from anything like isotropic distribution in angle, are distinguishing features of the nuclear cascade. The present experimental material is fully consistent with the model we have outlined here. Of course, the evaporation of low-energy particles, among them often complex nuclei of A from 2 to 10, with isotropic directional distribution, is expected and observed to follow, completing the de-excitation of the struck nucleus (Bll, H3). In both calculations reported, the neutrons and protons were not distinguished, but replaced by a gas of nucleons with about average properties. This reflects the feeling that the cascade particles will roughly be randomly divided into neutrons and protons, without much correlation between particle type and either direction or energy. The fact that in heavy nuclei, especially, there are more neutrons than pro- tons is perhaps the principal difference to be recalled, apart from the collision cross sections. The Coulomb barrier and the binding energy differences, especially the symmetry energy, which tends to keep the nucleus from becoming either proton- or neutron-rich in excess, will be unimportant factors in modifying the cascade, because of the typically Sec. 11B] Nuclear Reactions at High Energy 153 high energies of the emitted nucleons, in strong contrast to the evapora- tion mode of disintegration. Some detailed examination of the Monte Carlo results tended to verify the idea that nucleons emitted would be about half protons and half neutrons, for the nucleus with A = 100 and a 400-Mev incident proton. It is important to keep this feature of the cascade in mind when discussing the experimental data, which for the most part give information concerning only the ionizing prongs of the star event. There is ample empirical evidence, however, that the neutrons are in fact emitted (C8). The actual collision cross sections for free collisions are still of course not entirely certain. The empirical situation is discussed in Part IV, Volume I, but the principal question comes from the very curious ob- served behavior of a p . p , which is so strikingly independent of both energy and angle above about 100 Mev. Whether this property is one character- istic of the special states allowed by the exclusion principle in a proton- proton collision, or whether it reflects some deeper difference between the behavior of protons and neutrons, is a matter which will much affect the easy treatment of nucleons on an equal footing. It is this flatness of the cross section which is responsible for the increase of the proton energy loss in nuclei of medium size at high energies, and consequent growth in star size as the nucleon energy goes from 100 to 400 Mev, even in medium- A nuclei. B. Correlations among Nucleons Of course, the neglect of all nucleon interaction in the Fermi gas model is to be regarded as a zeroth approximation. In fact, the nucleons are not uniformly distributed either in momentum space or in coordinate space. The nuclear matter is lumpy, with constantly forming and dis- solving groups of nucleons. When a given nucleon enters into a collision, it is not free. Some momentum may be transferred to a neighbor through the force binding the two; or the recoil may be regarded as shared by a correlated spatial cluster of two or even more nucleons. In the complex process of the cascade, with high momentum transfers and repeated events, these effects show up only as eventual improvements to our present crude theory; but there are already known processes in which these correlation effects are, so to speak, the whole story. In the single elastic scattering of fast protons by light nuclei (where the cascade has no room to develop), in processes of typically low momentum transfer (as reactions induced by gamma-rays and by the absorption of slow mesons), and in the so-called "pick-up" processes (the inverse of deuteron stripping, in which a fast nucleon leaves the nucleus no longer alone but now paired with another nucleon which originated in the target, to 154 A Survey of Nuclear Reactions [Pt. VI form a stable fast-moving deuteron), the influence of the correlations and their consequent momentum distribution is decisive. The probability for finding two nucleons of the Fermi gas at positions r x and r 2 respectively is of course given by the integral ^(ri, r a ) = £ fdr 3 - ■ -dr N \ *( ri - • -r N ) | 2 ap J where the wave function >P is given by the determinantal expression ^ <Pb(ri)<f>b(r2) ■ ,<Pa(*j) =^e*"-"x(o->) Using the orthogonality of the individual plane waves for different allowed momenta hk a , and replacing the sum over values of k a by an y /•Pmax integral over momentum space £ ~~ * ; I dp, we obtain the k a (2irh) 3 J result P(ri,r 2 ) = 22 — 2 (Ufa) where the o- li2 give the spin, and t x , 2 the character, of the nucleon at points t\ and r 2 . The space function w depends only on the distance between the nucleons and is given by , | 3(sin 5 — 5 cos 5) . w(\ ri - r 2 |) = w(8) = h5 = p max | rj - r 2 | o (Hlb) Thus there is no correlation in position between unlike nucleons, or like nucleons differing in spin; but between nucleons alike in type and spin there is a definite spatial correlation arising from the exclusion principle, even if all forces between nucleons are neglected. Presence of interaction forces of course affects the wave function and hence the correlations. Using the Fermi gas as a zeroth approximation, calculations have been made to find what the spatial correlation is like under various assumed forces. Figure 28 shows the general character of the results. The momentum space distribution is given with and without interactions. The details of course depend on the forces assumed to act, but the gen- eral effect is to replace the rectangular Fermi gas momentum distribution with one which contains some components of higher momentum — traceable in the usual perturbation theory to transitions to unoccupied states of the Fermi gas spectrum induced by collisions. The actual distribution, with interactions, is somewhat as though the Fermi gas Sec. 11B] Nuclear Reactions at High Energy 155 were present but not at zero temperature. A calculation by Watanabe, assuming certain Gaussian potentials, indicated that the ground state of the nucleus of high A might correspond roughly to the momentum distribution in a Fermi gas of nucleons at a temperature of 6 or 8 Mev (W2). There are many processes, mentioned above, which reflect the mo- mentum distribution of the nucleons in the nucleus. One of the most Fermi gas Fig. 28. The momentum distribution of nucleons in a Fermi gas at zero excitation, and the same distribution when interactions are considered. Note that the effect of interactions in momentum space simulates the presence of thermal excitation. interesting is the rather unexpected process called "pick-up." Table 13 gives a summary of some experimental results which make it plain that the pick-up process is by no means a rare one (Y4). TABLE 13 Pick-Up Cross Sections fob 90-Mev Neutbons C Cu Pb "■inelastic, all events 0.2 0.8 1.8 barns Relative cross section for events yielding: Cascade protons with E p > 20 Mev 40% 30% 25% Pick-up deuterons {E d > 27 Mev) 12% 7% 4% Evidently the escape of the deuteron intact is relatively more difficult for the larger nuclei. It is to be expected that a deuteron can leak out of a good-sized nucleus at these energies only if it is formed in the far sur- face region; otherwise one or the other of the nucleons within it will collide in leaving the nucleus. This implies that the formation of such 156 A Survey of Nuclear Reactions [Pt. VI a system is really quite probable when the whole nucleus is taken into account. The process can be described in rather classical language as the en- counter of the incoming nucleon with another nucleon, of the right type and moving with the right direction and speed, so that the two move along together, forming a deuteron when the binding forces become effective. The whole event is by no means localized in a small region of space within the nucleus, since the deuteron is a big structure. Pre- sumably the nuclear edge contributes most of the pick-up processes, when nucleon passages outside the nuclear volume by distances as great as the range of the nuclear forces often contribute. Energy and mo- mentum must be pretty nearly conserved between the two particles, since a strong recoil would imply the break-up of the weak deuteron bond. The whole picture is roughly confirmed by experiment; deuterons do appear in fair number, coming off mostly forward, with about the energy of the incoming particle. At other angles the effect is smaller, and the typical energies lower. The incoming nucleon seeks a partner moving with the same momentum; then the two could move with negligible relative kinetic energy, and deuteron formation would be easy. But such a process would rob the nucleus of a great deal of energy, since essentially it would then emit two particles each with kinetic energy equal to that brought in by only one. The over-all energy and momentum conservation then implies emission of somewhat slower deuterons. The process is treated by an approximation that is physically very satisfactory. We make two assumptions: (1) The incoming neutron interacts with only a single proton at a time. All multiple events, sub- sequent collisions, etc., are neglected, to be treated separately. We will find the cross section for a single proton and simply multiply the result by Z. (2) During the collision, the forces between the collision partners are so strong that they overwhelm all the forces which bind the struck nucleon into the nucleus. The whole effect of the binding forces, which are by no means neglected, is thought of as determining the initial momentum distribution of the struck nucleon in the original nuclear ground state. For this second assumption, the approximation has been given the name of the impulse approximation; it has found application in a variety of problems involving collisions with a complex system (Cl). We write the usual form for the cross section per proton from the time-dependent perturbation theory: 2x . .„ d*df = — Pf\ H of | 2 dE n Sec. 1 IB] Nuclear Reactions at High Energy 157 Call the final deuteron momentum, measured in the center-of-mass system of the struck nucleus, hK; the initial neutron momentum hk . Write the energy of excitation of the residual nucleus W/ — Ef — E , and the deuteron binding energy B. If we normalize the continuum states of the initial neutron and the final deuteron center-of-mass motion to unit volume, we have for the cross section per proton, for deuteron emission into solid angle dfi leaving the nucleus with excitation Wf, 2tt ,__,„. 2*- M 2MhK , C 1 M 2 K ^=^1^0 = --.— |F 0/ N0 27T 2 h 4 k ■dSl\V of \ 2 (112) The matrix element | V f | is that of the neutron-proton interaction taken between the initial and final states normalized as described. These would be unit-amplitude plane waves for the incident neutron and the outgoing deuteron center-of-mass if we used the Born approximation, but the present calculation is not restricted to that simplification. For present purposes, we will simplify by neglecting the change of deuteron momentum with Wf, which is good in the limit of high neutron momen- tum, and take merely some average Wf. Now let us take as given the undisturbed wave function for the proton in its initial state, bound to the nucleus. Call this &(%), writing it as usual as a function of the proton position, r p . We can find from this by Fourier transformation the momentum-space wave function, "^-T^/^,*, (2x) where now | <pi | 2 is the probability for an initial proton momentum k p . The chief assumption of the impulse approximation now tells us that this initial momentum distribution alone affects the process, which in all other respects behaves as though the proton were initially free. Then over-all conservation requires the two conditions : (a) K = ko + k (113) h 2 X 2 M 2 _ (b) = B-\ + W f W 4M 2M We will also write the wave function for the neutron-proton system in the initial and the final states as fe, ^ respectively. Here each is a "deu- teron" wave function: in the initial state, the "deuteron" is of course 158 A Survey of Nuclear Reactions [Pt. VI highly excited, with a large positive energy, while in the final state the outgoing deuteron is bound, in its ground state, \f/ , of internal motion. These functions, like the interaction potential, we take to depend only on the relative coordinates of neutron and proton; the center-of-mass coordinates simply lead to the required momentum conservation. In the initial state the relative internal momentum of the neutron and proton is given by (k p — k )/2 = K/2 - k . The factor of one-half arises in the usual way from the introduction of the reduced mass in the equivalent one-body problem expressing the relative motion. Now the squared matrix element of the interaction, which determines the cross section, can be written, for a single proton, | H of | 2 = | w (kp) I 2 X | fdr+ *(r n - x p )V(x n - r p )fo(r re - r p ) (114) The content of the impulse approximation is just the writing down of this expression, first, to be summed over all the protons, and, second, as a simple product of the two-particle scattering matrix element with the probability for a given proton initial momentum. Now we can write the wave functions and scattering potential V(i n — r p ) in the momentum representation, giving them as a function of the relative momentum, say q, and the matrix element can be written (0 V \D) ^o(q), ^(q)^ »(f-kb) (115) Thus the scattering amplitude in this two-body collision determines the whole process, using (114). In principle, this amplitude could be ob- tained by a complete knowledge of neutron-proton scattering experi- mental values, though the most important range of q is not easily studied in this way. The energy conservation (113) can be used to fix | q |. A related calculation has been carried out numerically by Heidmann (H5). It is instructive to apply the Born approximation to our somewhat more general result. For simplicity let us neglect spin questions, and consider only the triplet states of the original collision, putting a factor of % into the cross-section formula (112). Then we will replace the initial neutron-proton wave function fe by a plane wave, giving simply a delta-function in momentum space. We can write for V just T n . p — E and the matrix element for the scattering, (0 | V | D), becomes (B + h 2 (K/2 - k ) 2 /M) X «,(K/2 - k ) The operator T — E has been transposed, so that it acts on the final Sec. 11C] Nuclear Reactions at High Energy 159 state wave function <p . With this done, we can write the entire quantity | H f | 2 , which we need to sum over all the protons, in the form D | H„ | 2 = Z\ w | 2 X (B + h 2 (K/2 - k ) 2 /M) 2 X | * (K/2 - k ) | 2 V (116) The first factor Z\ </>i\ 2 is simply the number of protons to be found with initial momentum K — k in the original nucleus; the second factor is the energy operator; and the third factor just the probability for a definite relative momentum q = K/2 — k in the deuteron ground state. With a reasonable choice of \f/ , say the familiar Hultheh (C2) approxi- mation, iAo(r) = const. (e~ ar — e~^ T )/r, it turns out that the last two factors roughly compensate, so that the variation of as with neutron en- ergy is dominated almost wholly by the proton momentum distribution. The cross sections given by the impulse approximation certainly account for the rather surprising frequency of pick-up deuterons, and for their distribution in energy and angle. Indeed, the calculations apparently overestimate the frequency of pick-up, presumably because subsequent interactions break up the nascent deuterons. But it is most satisfactory that this process, so unexpected at high energy, can, espe- cially in just that region, be given such a simple explanation in terms of momentum transfer. The pick-up process is the first of many found to lend themselves to the impulse treatment, and hence found to depend very much upon the momentum distribution within a nucleus : elastic scattering of nucleons, photon absorption, and many meson processes, like production by nucleons and gammas and capture by nuclei, are in this category (C3, F15). The description of the nuclear matter in momentum space, both by statistical and by more detailed methods, is sure to become more important and more familiar, supplementing the present description given most frequently in coordinate space only. C. The Optical Model for the Scattering of Nucleons At high energies, as indeed even at lower ones, the simplest of scatter- ing experiments is the measurement of the over-all attenuation of a beam of incident nucleons. If the experiment is done under conditions in which the energy is well-defined, and with so-called "good geometry" (i.e., an arrangement such that particles which have deviated even slightly from the original beam direction are excluded from detection), it is possible to measure also the fraction of the beam which has been elastically scattered. The convenience of such experiments has led to the use of a quite abstract and general nuclear model, which is tailor- 160 A Survey of Nuclear Reactions [Pt. VI made to give elastic and total cross sections without at all worrying about the details of the nuclear collisions. In an earlier section we saw how the assumption that the interaction between neutron or proton and nucleus was strong led to the idea of a sticky nucleus, opaque to the passage of nucleons, and then to a total cross section which for energies sufficiently high became just twice the geometrical cross section of the nucleus. The elastic scattering was represented by the portion of the emergent wave which produced the ordinary shadow behind the nuclear obstacle by interference with the incident wave. For a black obstacle we saw that this gave a coherent elastic, or "shadow scattering," cross section which was just equal to the inelastic cross section %R 2 . Its angular distribution was also determined. The optical model here used (F6) is a straightforward generalization of the results obtained earlier, extended to the case where the obstacle is no longer black but "gray," for the free path for nucleon collision in nuclear matter is no longer negligible, as we have seen above, in the hundred-Mev region. The ideas of physical optics, however, in the familiar Kirchhoff-Fraunhofer approximation, become better and better because %/R <3C 1. For this model, we think of the nucleus as a gray refracting sphere. Its opacity, or optical density, and its index of refraction for the Schrod- inger waves characterize the model. Then the amplitude of the scattered waves, and the damping due to absorption, can be computed, generally neglecting all interface reflections, and considering only "volume" effects, in analogy with the W.K.B. approximation. In its simplest form, the model regards all nuclei as uniform gray spheres, which differ only in diameter. This geometry is of course too simple, and in no way required by the optical approximation. It may be better to think of the nuclear matter more realistically as thinning out gradually toward the edge from a uniform central core; there is some evidence for such a model in exactly this sort of calculation. The nuclear matter is assigned a complex refractive index, N = n + it. Here n is the ratio of the nucleon wave number k at a given point within the nuclear volume to the wave number k Q = (2ME) 1A /h of the incident nucleon outside in free space (neglecting Coulomb forces). The quan- tity r specifies the amplitude damping within nuclear matter; the wave amplitude changes by a factor 1/e in a traversal of a distance within the nuclear matter equal to one wavelength of the incident particle, \(/(x) = io-e~ xk ° T , and is related to the mean free path for interaction in nuclear matter I — y^k^r. The inelastic scattering which we will calcu- late includes all exchange and absorption effects; it depends only upon the imaginary part of N, that is, on t. The real part of the index of refraction will determine the coherent scattering, in which the scattered Sec. 11C] Nuclear Reactions at High Energy 161 wave represents particles of the same type as those of the incident beam and having exactly the same energy in the center-of-mass system. We will consider only the uniform-sphere model; for this, N is constant within the nuclear sphere of radius R, and unity everywhere outside. In the shadow of the sphere, the transmitted wave which passes at a distance from the central diameter of the sphere (see Fig. 29) emerges with an amplitude a(p)e' koR , where e lk ° B is the amplitude of any portion of the incident plane wave which misses the sphere. The region of the shadow must contain a wave of lessened amplitude whose intensity Beam Source plane (a) (6) Image plane Fig. 29. The geometry of the optical model for nuclear scattering. In (a) the beam is shown incident on the nuclear sphere of radius R. The path length of a particular ray will clearly be 2s. In (b) the geometry of the diffraction calculation is shown. The source and image planes are normal to the incident beam direction. For each scattering angle 6 and azimuth * in the image plane, the total contribution is found by integrating the contributions of every radius and azimuth angle 4> in the source plane. This is the familiar Fraunhofer procedure. decrease gives the total incoherent absorption cross section, simply the total contribution to 1 — | a | 2 integrated over the whole area of the obstacle. To obtain the coherent, elastic, or diffraction scattering, one has instead to find the net amplitude which, added to the incident plane wave in the shadow region, will yield the transmitted amplitude a. This is plainly 1 — a, and gives a cross section | 1 — a | 2 . In the case of a spherical obstacle, the factor a(p) is evidently (j( n \ iko(B — 2s) (n+ir)2ikos Jko(n — 1) -2s — 2rkos (117) with s 2 = R 2 — p 2 . Integrating 1 — | a | 2 , we obtain the absorption cross section: tfabe = I Pdpi dp(l - e- iTSk ") = 2,r I s ds(l - e- 2 " 1 ) Jo Jo J [ 1 - (1 + 2R/l)e- 2R ' l l " ' B L 1 2RVP J (118) 162 A Survey of Nuclear Reactions [Pt. VI The elastic scattering is rather more complicated. It is given by the integral (F6) : •rdiff =fpd P f *d<p\ 1 - e -*<v<-2^-i)*o> p (n9) The angular distribution of the diffraction scattering can be obtained in a similar way. The approximation here is to sum the contributions reaching a given image point, far behind the sphere, from every source point in a plane wave front of amplitude a — 1 within the geometrical shadow. Then the Kirchhoff integral (of optics) can be written const R ffd<p p dp e ik ° R - pA B| X[1- a(s)] (120) where the geometrical relations are those of Fig. 29. Using polar co- ordinates in the effective source and image planes, we can write R-v i r = p(cos <p cos $ + sin <p sin $) X sin @ and the entire integral becomes "p dp d<p e ik ° 8in e cos ( *-*V(s) - 1] If Using a familiar representation for the Bessel function, we can write (F6) p dp J (k p sin @)[1 — a(s)] (121a) -J •'o and the differential scattering cross section dc/dil is just proportional to I «ac | 2 - We can evaluate the constant in the limiting case of R/l —+ <». For a totally black sphere, this goes over to the case already given in Eq. (46), namely: da/dti = R^J^koR sin 6)/sin @] 2 . For the gray spheres here under consideration, the evaluation can be carried out. By replacing the integral over p by the sum over I, in which I + ^ = kp and the relation Jq[(1 + 2) s * n ®1 = Pi( cos ©)> valid for large I and small @, is employed, we find l + V 2 <kB a sc ~ E (21 + l)Pi(cos »)[1 - a(si)] with si = [(k R) 2 -(1 + \fV A IK (121b) This is very reminiscent of the partial-wave analysis of scattering, and indeed is just the W.K.B. approximation to the phase shifts of the Rayleigh partial wave procedure. That procedure has been used ex- Sec. 11C] Nuclear Reactions at High Energy 163 actly (PI), giving a result somewhat larger than does (121b) at small angles of scattering. A series procedure is also applicable for the evalua- tion of (121a) directly. The results of such a calculation are indicated in Fig. 30. The effect of nuclear transparency is to reduce the contribution of the shadow scattering as the transparency increases (or R/l decreases). The graphs show this reduction in the case of a plausible value for the index of re- fraction, implying some increase in phase velocity within the nucleus. 2.0 - 03 J= 1.0 ^ M=1.16 "totA^ 2 <W*--R 2 °abs/Tfl 2 1.0 R/l 2.0 Fig. 30. Cross sections computed from the optical model. The diffraction, absorp- tion, and total cross sections, measured in units of the geometrical cross section, are given as a function of the nuclear radius measured in units of the free path. A particular value, n = 1.16, has been chosen for the refractive index. From (F5). When n is not simply unity, the diffraction scattering may considerably exceed the geometrical limit wR 2 for a black nucleus. By fitting the data on neutron scattering at 90 Mev to the results of Eqs. (118) and (119), we can choose n + ir in such a way that the value of R obtained for a certain choice of n + ir fits for all nuclei from Li to U the simple relation R = r A H , with r chosen at 1.39 X 10 -13 cm, a very reasonable value. Suitable values, which are consistent with direct scattering experiments for neutron-proton and proton-proton scattering at this energy are I = 3.3 X 10 -13 cm and n = 1.16 (F5). For these values the coherent scattering is larger than the geometrical value. A less empirical evaluation of n + ir is possible in two ways. The nuclear material simply acts to change the phase velocity of the waves, and can thus be represented by a potential well of depth V, with the relation n = k/k = (1 + V/E) y% . This works well enough for 90-Mev neutrons, but the same choice by no means satisfies the experiments at higher energy, where the variation of total cross section with energy does not follow the predictions of the simple potential well idea. A much more sophisticated approach has been made (J2) by means of the 164 A Survey of Nuclear Reactions [Pt. VI familiar result of optical dispersion theory, which gives the refractive index in terms of the scattering amplitudes for forward scattering by the individual nucleons. The relation is n - 1 = K-„(0) + a n . p (0)]Trp/k 2 (122) where p is the density of nucleons in the nucleus, and the a's are the appropriate scattering amplitudes. Use of the empirical cross sections da/dil barns 24° 16° 8° 0° Neutron scattering angle Fig. 31. Angular distributions of elastic scattering computed from the optical model. The differential cross section is shown as a function of scattering angle for two nuclear radii (corresponding to copper and to lead). The experimental points are shown. All this is for 90-Mev neutrons incident. From (F5) and (CIO). is required for I, .but it is of course not enough to fix the quantities a in (122) unambiguously. Best fit with the observed total cross sections for a nucleus like Al is found when the real part of the index of refraction is made to go rapidly to unity, k/k — > 1 — as the energy increases above 100 Mev. Such a scattering amplitude results from a nucleon-nucleon potential consisting of a strong repulsive core surrounded by an attrac- tive well, with a core radius of some 10 -13 cm. This result can be taken as some confirmation of the view, based on the very flat and strongly isotropic proton-proton cross section above 100 Mev, that some such "hard-core" model of the nucleon-nucleon interaction is at least phe- nomenologically suitable. With a n . v = a p . p the fit for the total cross section to relation (119) is quite good. The angular distributions are another source of information. In Fig. 31 we show a comparison between theory and experiment for neutrons Sec. 11D] Nuclear Reactions at High Energy 165 of about 90 Mev. The diffraction pattern is qualitatively correct, but the experimental peaks are a little sharper, as though the nucleus were somewhat smaller than expected from the constant-density model. Experiments at higher energy show the positions of the successive minima in the diffraction pattern, and indicate that the minima are in fact blurred out, perhaps only slightly in the heaviest nuclei, but beyond recognition in the lightest ones. This is consistent with the results to be expected if we surrender the naive idea of a strictly uniform sphere of nuclear matter and imagine instead that the nucleus has a core of con- stant density surrounded by a fringe of nuclear matter with a thickness something like the range of nuclear forces within which the density falls gradually to zero. Such a model is of course actually implied by the simple notion of a nuclear surface, as it is for any classical drop of liquid. Taking such a model, and the connection between n and the empirical nucleon-nucleon cross sections given by the dispersion theory, Jastrow (J3) claims a reasonable fit, both to the total cross sections and to the details of the angular distribution of the diffraction scattering, over a wide range of energy and nuclear mass. Much more remains to be done. D. Processes of Nuclear De-excitation at High Energy Let us return to the topic first mentioned in this chapter: the detailed course of nuclear disintegration in the region of hundred-Mev energies. In the step-by-step treatment of the cascade, each particle was followed until the kinetic energy it possessed, relative to the bottom of the over- all nuclear potential well, fell below a certain limit, some 30 Mev. After this, no single collision between this particle and the nuclear edge is likely to result in the emission of the particle; rather, reflection will occur, and the energy will gradually distribute over the whole of the nuclear system. Not until it is again by chance concentrated on a single nucleon (or in a single well-defined mode of nuclear motion) will emission occur. This is the basis of the compound nucleus treatment which is the main burden of this entire discussion. To place matters in terms of the familiar thermal analogy: excitation of a nucleus by collision with a single nucleon, or a small nucleus, corresponds to heating the nuclear matter very hot indeed in a small spatial region. From this region a nucleon or several may emerge, taking off most of the energy. But the heat will proceed to spread. The time of conduction of energy over the whole, compared with the time it takes a single nucleon or a few nucleons of a cascade to emerge, will determine whether or not the subse- quent steps will proceed by thermal evaporation. Our treatment assumes that, after a certain energy loss has been suffered by each fast- 166 A Survey of Nuclear Reactions [Pt. VI recoiling nucleon, all the remaining excitation energy passes into thermal form. Since the spread of energy is mainly through the zero-point motion of the Fermi gas of nucleons, this implies that the statistical treatment will be valuable when the individual nucleon has an excitation not large compared to its zero-point energy, well under 20 Mev or so. This limits the process plainly to cases where the whole nucleus is excited by a good deal less than its total binding energy, say by some 5A Mev at the most. We have already seen that such excitations are ample to handle the most frequent cases observed in the bombardments with Bi(p, 3n) 10 20 Energy of incident protons (Mev) Fig. 32. Reaction cross sections for various proton-induced reactions on bismuth. The competition between the several modes of decay is shown strikingly here; as each new mode becomes energetically possible, it tends to drain away the previously favored decay method. After (K5). nucleons up to almost 500 Mev in fairly heavy targets, and they cover also, at least in not too small nuclei, the interesting phenomena following upon meson absorption, both w and n (B12, M6). The statistical treatment we have used earlier in the low-energy cases implies of course a thermal equilibrium, which will not actually exist in the case of high excitation and the emission of many particles. We can expect that a kind of transient equilibrium will in most cases exist between the successive nucleon emissions which make up the total event. The limit on such an idea is no different from that mentioned just above: the time for rediffusion of the excitation energy after an act of emission, governed by the thermal transport, should be small compared to the delay before the next concentration of energy on a single emitted particle. The typical thermal transport time is under 10 -21 sec; while the time for nucleon emission up to the highest excitations here considered is ten thousand times as great. The whole division of the process into an immediate cascade and a subsequent evaporation is of course approxi- Sec. 11D] Nuclear Reactions at High Energy 167 mate; but within these limits the approximation is successful to a rather high degree, and very instructive. Of course, the transition from the ordinary statistical treatment of reactions like (n,a) or (n,p) to the large evaporative stars so dramatically revealed in the nuclear emulsion studies is a continuous one. For in- stance, an elegant series of experiments by Kelly (K6) has carried the simple competition theory of formation of the compound nucleus and its successive decay (see Section 7) through the whole series of reactions with the target nucleus Bi 209 : (p,n), (p,2ri), (p,3n), (p,4n). Excitation energies in this experiment ranged up to about 40 Mev. Figure 32 shows the results obtained, which fit the expected relations of the statistical theory of competition [see Eq. (89)] very well indeed. If we write the ratio of excess energy of excitation over threshold to the temperature as x, the data fit the simple result a{p,2n)/a v = 1 — (1 + x)e~ x very well by adjusting the single parameter, the temperature kT. The value chosen for kT agrees completely with the calculated value for the semi- empirical level density formula, Eq. (31), about 1.1 Mev. 1. Spallation and Evaporation Stars. As the bombarding, and hence in general the excitation energy, continues to rise, we reach a region of higher complexity. Here not only a single product, or a few related competitors, can emerge, but a whole series of successive steps can be taken, each one in many alternative ways. This fact, and the often important fluctuations away from the mean statistical behavior of even such a complex multiple evaporation, is the reason for the absence of any well-codified study of this general subject. So far we can give fairly satisfactory but rather tedious means of calculation, but no compact summary of the possible results. Two different experimental approaches to the field have led to different nomenclature, and to somewhat different descriptions, of one and the same phenomenon. The radiochemical procedures of separating the various radioactive products from targets bombarded with projectiles at differing energies, and so obtaining yield curves for a whole series of possible products, is one powerful means of study (B4). Here of course only the over-all change is observed; if A and Z change by 20 and 10 say, the event can be described as the emission of 10 neutrons and 10 protons. But the emission of alphas or even other small nuclear fragments in such an event is by no means excluded. Typically a high- energy bombardment yields a bewilderingly large number of product nuclear species. By extension from the idea of the fission of a nucleus into two large fragments, with a little neutron spray, the radiochemists have come to call these processes spallation, with the implication of the 168 A Survey of Nuclear Reactions [Pt. VI emission of a whole series of small fragments, rather than the cleavage into two big ones familiar in fission. On the other hand, the cosmic-ray workers, first with their cloud cham- bers (P4), and more recently with the powerful nuclear emulsion tech- niques (B9, H3), have typically studied not the statistical residue of millions of disintegrations by identification of the product nuclei, but the ionizing fragments released from each individual nuclear break-up, with the familiar bristle of ionized rays, each marking the trajectory of a single ionizing fragment starting in the central decaying nucleus and ending at the end of the fragment's range. In this method, besides the simple count of the fragments and charge carried off, the energy and angular distribution can also be obtained. But a good statistical sample is tedious to collect, and neutrons are invariably missed. (A few ex- periments with counters have satisfied us that the invisible neutrons, expected in more or less equal numbers with the visible "prongs," are really emitted.) The typical appearance of the event in the emulsion has led to its designation as a "nuclear star." Star or spallation reaction, the event is the same, but the method of detection, and to some extent the features used for description, differ widely. In Fig. 33 we present a rather generalized summary of the results of a typical spallation study: the bombardment of a copper target with protons of 340 Mev (B4). The contours in the N,Z plane show the ob- served yields, with some plausible extrapolations and smoothings. Characteristic is the large number of observed nuclear species, more than thirty having been identified. The biggest yield is for nuclei differing by only a few units in Z and N, plainly due to events in which not much excitation was left behind after a small cascade. There is a strong tendency to stay in the vicinity of the stable valley; it is not likely that the neutron-proton balance will be badly upset by evaporative events in which nearly all the emitted nucleons are protons (or neutrons). This tendency is compelled by the quasi-equilibrium theory, and favored by the energetically advantageous emission of alpha-particles themselves. The spallation studies have shown one other rather interesting phe- nomenon (B5). Both by identification of the short-lived isotope Li 8 , and by arguments of energy conservation, it has been demonstrated that at least a few events are possible which bridge the gap between simple alpha-emission and straightforward fission. In Section 10, it was pointed out that the splitting-off of small fragments is energetically favorable for a wide range of A because of the balance of Coulomb and surface energy. In spallation experiments, where infrequent events can be detected with relative ease, it has been found that there is indeed a small yield of nuclei of mass far removed from that of the target even Sec. 11D] Nuclear Reactions at High Energy 169 for rather low bombardment energies. For example, 70-Mev protons on Cu give a small yield of CI 38 . The connecting reaction which in- volves emission of single nucleons and alphas only is the reaction Cu 63 (p,6apn)Cl 38 . But that reaction has a threshold, calculated from mass differences, of some 110 Mev. The extreme reaction, from the Sq25 - Fig. 33. A contour map plotting the yield of a spallation reaction in the N,Z plane. The target nuclei are marked with solid circles; copper was bombarded with 340- Mev protons (B5). The yields center roughly on the line of greatest stability, shown dashed. point of view of minimizing surface energy, is the three-particle break-up, Cu 63 (p,nAl 25 )Cl 38 . The threshold for this rather unlikely course of reaction is about 50 Mev. It thus seems established that a variety of heavier fragments, of which Li 8 has surely been found, but probably extending over a wide range of masses, up to processes splitting the nucleus into several pieces of approximately equal weight, can in fact take place. This marks a kind of transition to the case of low-energy fission, and implies that the statistical competition of specialized vibra- tions of the nucleus leading to massive splits will have to be taken into account in a complete theory of cooling. 170 A Survey of Nuclear Reactions [Pt. VI 2. Evaporative Stars: The Fundamental Process. The study of the energy and angular distribution of observed star prongs enables a rough distinction — allowing considerable overlap — between the cascade and its evaporative consequences. The evaporative prongs are gen- erally marked by: (1) a roughly isotropic angular distribution, (2) a rapid decrease in prong frequency as energy increases. The long tail up to high energies actually observed — so-called "gray" and "sparse black" prongs, named from their low ionization densities in the emulsion (B9) — are regarded as cascade particles. True evaporative particles ought to show a typically quasi-Maxwellian drop-off of number with energy, and no such long tail to energies far beyond any possible mean particle energy. In general, we expect the evaporative process to be sensitive, especially in its last stages, to details of nuclear structure, like the odd-even property and even more refined shell-like behavior. The thermodynamic treatment of the stars is based on the statistical formula, due to Weisskopf, which connects the probability for dis- integration of a compound nucleus with the formation cross section, by the principle of detailed balance. [Compare the discussion in Section 5, Eqs. (51) and (52).] We write gM p(f) 2 h 3 p(i) where P(T) dT is the probability per unit time for emission of a particle of mass M and spin statistical weight g with kinetic energy in dT; a is the cross section for capture of the same particle by the residual nucleus / to form the initial nucleus i; and the functions p are the level densities of initial and final nucleus as indicated, functions of course of the mass and charge number and of the initial excitation energy, W, and final excitation W. In principle, we may now use the function Pj(T) for emission of a particle of type j with mass A 3 - and charge Z } - from the initial nucleus of given kind and given excitation energy, W = W + T + Bj, with B } - the binding energy, and then simply follow the course of the successive emissions until all the excitation energy has been converted into kinetic and potential energy of the fragments, and into radiation. This implies the solution of the set of coupled equations which can be given, if only formally, as (F13) dy(Z,A,W,t) „_ r x ' = E dW'y(Z + Z h A + AjWdPjiW'.W' - W) dt j JW+Bj P(T)dT = ^—- ff (T)T — dT (123) - Z y(Z,A,W,t) f P 3 {W,T) dT (124a) Sec. 1 ID] Nuclear Reactions at High Energy 171 Here the yield function y(Z,A,W,t) gives the probability of finding a nucleus with charge Z, mass A, excitation energy W, at time t after the start of the process, which fixes the initial condition: y(Z,A,W,t) = Saa, 5zz„ S(W - Wo) (124b) Now from the function y one can learn all the facts about the star, including of course both the mean behavior and fluctuations from it. For example, the total yield of a definite nuclear species is just given by Vtot(4,Z) = f 'y(Z,A,W,t -» oo) dW Naturally such a complete treatment is next to impossible; indeed, fluctuations are generally treated very casually [but see (Tl)]. We shall examine a more manageable approximation; perhaps the complete treatment will one day be performed by Monte Carlo technique, which seems very well suited to the task. 3. The Mean Behavior in Cooling. The most satisfactory approach is the study of the mean behavior of the evaporative process. To make this possible in general (F13), several approximations must be made, not all of easily controlled accuracy: 1. In the fundamental formula (123), the cross sections are replaced by very simple forms: usually just the geometrical irR 2 for neutron emis- sion, and a simplified expression like <r = wR 2 (l — V'/T) for a charged particle with a Coulomb barrier height given by V = 0.7(ZZje 2 /R), to take care of penetration effects. 2. The level density is expressed by some one of the results of a par- ticular nuclear model, discussed in Section 3. The detail with which the shell properties are represented may vary greatly. 3. The cooling proceeds usually by the emission of particles whose Aj and Zj are small compared to those of the initial and residual nucleus. This allows series expansion of functions of A and Z, performed as though A and Z were continuous variables, instead of integers. First of all we define the relative width (hence probability) for emitting a particle of a given kind, indexed by j. In the typical calculation, it may be worth while to consider half-a-dozen or more types of emitted particle, as proton, neutron, deuteron, H 3 or triton, He 3 , He 4 or alpha- particle, and sometimes even heavier fragments. In general, then, the mean emission width is given by the expression Tj(W) = f Pj(W,T) dT JVf+Bj where Pj is the transition rate given by (123). The approximation of 172 A Survey of Nuclear Reactions [Pt. VI mean cooling is of course just to consider the process as going always proportional to these mean emission rates, and to neglect the different courses of the evaporation which come from the fluctuations about the mean energy and type of decay. Naturally no single act follows the average behavior, and the successive steps in fact depend in turn upon just what particle and energy loss preceded them. These refinements are here neglected. Now we can calculate the mean energy of excitation lost by evaporation of particle j: X Pj(T)TdT 8Wj = Tj + Bj = + Bi (125a) X PjdT and we can express the mean cooling in one way by writing the mean rate of energy loss per nucleon emitted : — Z IV HWj SW j ■=- = (125b) 8A Sr,- 5Aj In a similar way we can write the mean number of particles of type j lost per unit energy loss : Srij Tj = = = (125c) 8W HTjdWj and for the entire cooling from initial to final excitation energy get — r w * Tj{W) dW dnj = | ^ (125d) ■ _ r w < Tj{W) dW ' 3 Jvj'+bj ]r Tj s~Wj Here there is of course one such equation for each of the values of j, say six or more, and the equations are all coupled. The main coupling is through the effect of the changing neutron-proton ratio, and some reasonably trustworthy solutions have been obtained (neglecting other relations) (L3). (a) Cooling Behavior with an Explicit Model. To gain any insight into the expected behavior of this phenomenon, it is necessary to intro- duce an explicit nuclear model which can fix the level density and its variation with Z,A, excitation energy, and even finer details like the odd-even effects. In Section 3 we set out a number of expressions from various models for the level density p(W). We choose here to use the rather simple model of the Fermi gas, which has some weak experimental Sec. 11D] Nuclear Reactions at High Energy 173 support, and can be regarded as the empirical expression of the results for nuclei of middle weight and for energies in the range here studied. With a Fermi energy of 22 Mev, we obtain [compare Eq. (31)] ^(.A.Z.W) (AW)** P (W) = v and S = 0.63 (126) P ^ ; t(2t dW/dr)* Mev Here the entropy is given as a function of the excitation energy W, and we have introduced the temperature t, writing W/Mev = At 2 /10, and t is in Mev. With this model the rate of evaporation becomes, neglecting some slowly varying ratios, P(T) dT = gM/ir 2 h 3 (T - V')e^~ Si) dT (127) and, if we make the approximation, valid for small emitted fragments and 8W « W,, of expanding the function S(A,W), we get S(A — SAj, W - 8W) = S A (W) - SW/t - SAjt/10 and ~(T - V) P{T) dT = const (T - V) exp (128a) T This is the familiar Maxwellian distribution of evaporated fragment energies. The mean kinetic energy taken off by an evaporated particle i is I x 2 e x dx Tj = t — = 2r (128b) 3 ,00 xe~ x dx I and the total width T 3 {W) for emission of j-particles with any energy T is approximately Tj = h P(t,W) dW JBj + Y' ~ a — i ■ exp I 1 t in Mev (129) ~ y 2x 11 Mev V 10 t / Both terms in the exponent here depend strongly on the type of particle emitted; they are the effective potential barrier V/ and the binding energy Bj. Now we must explicitly give the variation of binding energy with A and Z, which will of course determine the nature of the most probable emitted particle. We write a semi-empirical form for the binding energy of a nucleus in its ground state [compare Eq. 174 A Survey of Nuclear Reactions [Pt. VI (101)], with <(JV - Z)/A ) written for the value of (N - Z)/A which corresponds to the most stable nucleus of a given A : B -= const + Cl A - c 2 A[(N - Z)/A - { (N - Z)/A >] 2 Here we can take c x about 8.6 Mev, and c 2 about 23 Mev. These num- bers are suited for target nuclei in the middle of the table, A ~ 100, as in photographic emulsion. Now, if a particle of type j is evaporated, with 8n,j neutrons and SZj protons, taking off kinetic energy T, the drop in excitation is given by differentiating B, and regrouping the terms, to yield 8W = T + Bj [N - Z IN - Z\l Bj = Cl dAj - 2 |— \~-J~ / J < c " bn * ~ c * SZ ^ + 7 > f a IN - Z\ l\N-Z IN - Z\l) c " =C2 { 1 - A M\-i-/-i[-x- + \-x-)JH a8c2 f a IN - Z\ 1VN-Z IN - z\-\) (130) and Ij is the internal binding energy of the emitted fragment. The energy Bj is of course the threshold energy for emission of the given fragment with zero kinetic energy, which we call the binding energy of the fragment into the initial nucleus. We have considered only nuclei rather near the stable valley — (N — Z)/A ~ {{N — Z)/A ) — in evalu- ating c n and c z . Now we can examine at least the general cooling behavior by simply looking at the competition between the various emitted fragments. Q>) The Competition in Cooling. Let us look at the ratios of the r,- for the various particles. First, note that the statistical and mass factors g 5Aj alone will give relative weights for the six most common products as follows: p:n:d:T:He 3 :a: :1:1:3:3:3:2. Note the increase in statis- tical probability, especially for the heavy isotopes of hydrogen, which feel a low Coulomb barrier as well. Much more important, however, than the statistical and mass factors is the exponent in Eq. (129). We write here the expected emission width relative to that for neutron emission, using the energy relations of (130) . We can write the effective Coulomb barrier heights for nuclei of the emulsion (A ~ 90) as V/ = 6bj Mev, where the values of bj are esti- mated to include charge, radius, and mass effects on penetrability, and Sec. 11D] Nuclear Reactions at High Energy 175 we get b n = 0; b p = 0.7; b^ = 0.8 = br; &He 3 = 1-6. Then the widths, relative to the width for neutron emission, become r„/r„ = l iyr„ = exp [- (92(i. -v) + 4)/r] T d /V n = 3 exp [-t/10 - (56(i- - v) + ll)/r] T T /T n = 3 exp [-2t/10 - (18^ - v) + 14) /r] r H ea/r n = 3 exp [-2t/10 - (110(? - r) + 19)/t] TjT n = 2 exp [-3t/10 - (74(* - p) + 8)/r] The energies involved and the temperature r are all given in Mev. We have written v for the quantity (N — Z)/A ; the value of v for the most stable nucleus of a given A is written as V. Thus {v — v)A = neutron excess — proton excess. An examination of these ratios alone enables a qualitative discussion of the course of the mean curve of cooling. The most noteworthy features are these: 1. The Coulomb barrier — contributing to the last term of the ex- ponent — of course favors the emission of neutrons over charged particles, and reduces the emission of particles of charge +2, as long as it is of any consequence. Since the temperature r is ~(W)**, the effect of the Coulomb barrier on charged-particle emission will be decisive for all excitations up to some critical energy. For higher energies the emission of ions and neutrons will show little difference ascribable to Coulomb effects. The critical energy for proton barrier effect in these middle nuclei is an excitation of from 100 to 150 Mev; for helium isotopes, from 250 to 350 Mev. 2. Apart from the Coulomb barrier, the most striking term in the exponent of the relative emission widths is the symmetry energy, propor- tional to the fractional neutron excess of the initial nucleus, (v — v). Looking, for example, at the proton-neutron ratio, T p /T n , we see that this term favors neutron emission when there are too many neutrons compared to the region of stability, and favors proton emission when the nucleus is neutron deficient. This term has been called a "governor" term (L3), because it prevents great excursions from the region of stable nuclei in the cooling process. It arises of course from the parabolic form of the familiar Heisenberg valley in the nuclear energy surface. In many-particle stars it will more or less insure that more neutrons than protons are emitted. 3. The leading term in the exponent of the relative widths for emission of the complex star fragments — d, a, etc. — grows more negative as the 176 A Survey of Nuclear Reactions [Pt. VI excitation, and with it the temperature, increases. This term arises from the dependence, not of the energy, but of the entropy itself, on the number of nucleons in the nucleus. It favors the emission of lighter rather than heavier particles, because more ways exist of assembling a nucleus if it contains more nucleons. If it were not for this type of term, the emission of the heavier fragments would become more prob- able than that of single nucleons because of the purely statistical weights. Fig. 34. The course of the neutron number of an evaporating nucleus as a function of nuclear temperature. The neutron excess is plotted; it is the difference between the number of neutrons in the nucleus, N, and the number of neutrons characteristic of the most stable nucleus of the same mass number A. The neutron excess first increases and then decreases as the nucleus cools. The terminal value of N — (iV) is about —1, implying the subsequent emission of a slow proton, or simply /3 + -decay, following the main process of evaporation. See pages 174-177. The curves are drawn for a nucleus with initial A = 100, and for two initial temperatures, 4 and 8 Mev. After (L3). For example, without this term r a /T n — > 2 as t grows without limit. As star size increased, then, alpha-emission would outweigh proton emission and even neutron emission, a very odd result, and far from what is actually observed. With this entropy term, however, the ratio Ta/Tn goes through a very flat maximum in the region of 100-Mev excitation. As we have seen, however, in earlier chapters, the variation of level density with A is quite poorly known, and detailed predictions from this simple entropy assumption cannot be expected to agree in detail with experience. Now we can integrate the simultaneous equations (125) to give the full course of the cooling. This has been done in rather rough approxima- tions, perhaps best by LeCouteur, whose procedure we follow (L3). Sec. 11D] Nuclear Reactions at High Energy 177 In general, we can expect this typical state of affairs : the highly excited nucleus, say with 400 Mev, begins by evaporating neutrons and protons about equally. To begin, it has v — v almost zero — perhaps one or two protons and neutrons have been knocked out of the stable target nucleus in the nuclear cascade. But now the nucleus boils off several protons 200 400 600 Initial excitation energy (Mev) Fig. 35. The number of evaporated particles of several types as a function of initial excitation energy, for an evaporating nucleus of initial A = 100. The neutron num- ber scale is at the left; that for charged-particle emission at the right. After (L3). and several neutrons. By the time it has emitted say four of each, the cooling nucleus has become neutron-rich, with respect to the smoothed- over line of maximum stability, and the "governor" term begins to favor the loss of neutrons. Then the cooling proceeds, somewhat prefer- entially losing neutrons, until the excitation has dropped below some 100 Mev. From here on, proton loss will be almost prohibited by virtue of the Coulomb barrier, and the excitation energy will fall until no more particles can be emitted. The resultant nucleus is now neutron-deficient by a little less than one neutron on the average, more or less independent of the initial energy of excitation for not too low excitations. We show in Fig. 34 a schematic plot of the neutron number against excitation during the cooling, which follows the course here described. 178 A Survey of Nuclear Reactions [Pt. VI In Fig. 35, we plot, after LeCouteur, the mean numbers of various emitted particles for different excitation energies, with an initial nucleus of A — 100. The virtual independence of the relative yields of various particles upon energy is a feature of the theory; but the calculations have actually been done neglecting the small variations actually predicted, 10 20 30 Kinetic energy of evaporated particle (Mev) Fig. 36. Energy spectrum of evaporated particles for several initial excitation energies. Initial nucleus of A = 100. The solid curves all refer to emission of pro- tons; the dotted curve, to that of alpha-particles. After (L3). and using the relative yields given by equations for an initial excitation of 400 Mev. In Fig. 36 we give the expected kinetic energy distribution of evaporated fragments; it is given by the rough averaged formula, very much as in (127), by P(T) AT = V -(T-V)/t dT (132) Results are given for several excitation energies, and for both singly charged and doubly charged star prongs. Sec. 11D] Nuclear Reactions at High Energy 179 All these results are likely to be acceptable only for fairly large excita- tions (W > 150 Mev), and the theory cannot be taken seriously for A much less than about 60. The actual figures are for a definite A = 100. (c) The End of the Cooling Process: Low-Energy Prongs. We followed the cooling process down to the point where the temperature is so low that no further nucleon emission is possible energetically. But this threshold of course varies with the type of particle. Moreover, the process has left us with a nucleus which is neutron-poor by about 0.8 neutron. Then the values of the threshold energy are not the same as for a stable nucleus, and become, from (130), B p c^7 Mev, B n ~ 10 Mev, 5 a ~ 5 Mev for the three main possibilities. Thus, even though the Coulomb barrier reduces the rate of charged-particle emission for such low energies very much indeed, the neutron cannot compete at all, since the binding energy of the neutron in the neutron-poor residual nucleus is higher than normal. De-excitation can go on by gamma- emission, beta-decay, or charged-particle emission. The anticipated lifetime for these slow processes at an excitation of 10 Mev is perhaps 10~ 16 sec or so for the gamma, and some 10 -2 sector the beta. Protons emitted through the barrier with kinetic energies of anything above 0.5 Mev will favorably compete with such slow radiative processes. Alphas and other fragments of mass above one unit will experience a much greater loss in penetrability; since this factor is already 10 -10 , protons are about the only practical competitors. Thus, if a nuclear particle of any kind, the next-to-the-last to be emitted, leaves the residual nucleus with an excitation of less than 10 Mev and of more than B p + 0.5 or some 7 Mev, a proton will be emitted. If the remaining excitation is below B p , gamma-rays alone can come off. Thus about 30 percent of all nuclei left with 10 Mev excitation will emit slow protons, far more than would be expected if neutron competition were not excluded by the shift in binding energies due to the neutron deficit. A computation (L3) has shown that there is to be expected in emulsion nuclei about 0.2 proton per star with kinetic energies in the range from 0.5 to ~3 Mev. These very slow protons have apparently been observed (H3). In addition there is a considerable excess of slow alphas observed; this cannot be accounted for by such a mechanism, but is ascribed to a lowering of the effective barrier by processes to be considered in Section HE (L3). It will not escape the reader that this type of calculation has over- looked even the most striking features of nuclear shell structure, like the odd-even variation in energy content. This does not affect the previous phenomenon, slow-proton emission, but in general it should have some detailed effects on the closing scenes of the evaporation, when excitations are not large compared with the few Mev which represent 180 A Survey of Nuclear Reactions [Pt. VI shell-closing energies, etc. Moreover, the rather large gamma-ray width which seems to be observed in the neighborhood of 20 Mev excitation may occasionally compete — say one time in ten or a hundred — with the last one or two emitted particles. All these interesting but somewhat fine points await both more systematic experiment and a better theoreti- cal treatment. (d) Effects of High Nuclear Temperature. The calculations of evapora- tion ought not to neglect the changes in the nuclear properties which arise from the high nuclear temperatures during the earlier stages of the process. In equation (126) and in the binding-energy formulas of (130) we have over-simply considered the nuclei throughout as though they were in their ground state. The work of LeCouteur (see Figs. 34 and 35 which are essentially taken from his work) actually did not leave out this important and complicated circumstance. The effects to be considered are three in number, all closely related to thermal expansion: 1. Depression of the Coulomb barrier, and consequent favoring of emission of low-energy charged particles, as a result of the thermal expansion of the nucleus. An estimate of the thermal expansion of the nucleus due to an actual change in the equilibrium volume energy at high temperature, entirely analogous to the familiar phenomenon on large scale, can be obtained by thermodynamic arguments. We can write the internal energy of a nucleus of given A as U(V,t), showing its dependence on temperature and volume. For a definite temperature — actually t = — this internal energy is a minimum at some volume, say V . Expanding, we can write dU U(r,V) = U(t,V ) + — (V-V ) 2 +- But the coefficient of the linear term is zero by the definition of V - Now the Helmholtz free energy, for negligible external pressure, is just F = U - rS = U(r,V ) - tS(t,V) + - (v - v y ld 2 U 2~dV 2 and the actual equilibrium volume at any temperature r will satisfy = fixing the free energy at a minimum. We have dF the relation — ■ dV already given the entropy in terms of A (Eq. 126) and we can estimate the entropy as a function of volume by using the relation V = ^irr 3 -A. This of course is not precise; the entropy depends both on A explicitly, the number of nucleons, and on the volume for fixed A, but it will serve Sec. 11D] Nuclear Reactions at High Energy 181 for an estimate. Now we get S = 0.2At = 0.2 • F/(4Trr 3 /3) • t, and this yields an expression for the volume change with temperature: V = V + [0.2/(W/3)] • [r 2 /(d 2 U/dV 2 \ TjVo )] (133) The familiar thermodynamic result dF = —S dr — p dV for small depar- tures from equilibrium yields the relation dF/dF| T = —p. But the ordi- nary isothermal compressibility is defined as k = —(1/V) (dV/dp)\ T . Since dp/dV\ T = l/(dV/dp)\ T , we have l/« = Vd 2 F/dV 2 \ T . For low temperatures we can write, closely enough, d 2 F/3V 2 | T = d 2 U/dV 2 \ T . An estimate of the nuclear compressibility, d 2 U/dV 2 , is easy to obtain in several ways. An application of the virial theorem (78) gives d 2 U/dV 2 = fc£7 kin /F 2 , where the factor k is about 2; direct calcu- lations with models using a Fermi gas and correcting for assumed two- body interaction potentials are in fair accord. Some insight may be had by recalling that the velocity of sound v is given by the relation w 2 = -[V(d 2 U/dV 2 )]/p. If the sound velocity is estimated very roughly as simply the rms velocity of the particles in the Fermi gas, we have v 2 /c 2 = 6E F /5Mc 2 . Using (133), we can write the relative change in radius upon heating [R(t) - R(0)]/R(0) ^ %(r 2 /E F ) = 0.008t 2 (t in Mev), a small but not entirely negligible change. Bagge (Bl) has pointed out that there is likely to be another, more dynamical, source of reduction of the Coulomb barrier than this uniform volume expansion. Surface vibrations in the excited nuclear drop "wrinkle" the surface; over many cycles, protons in the nuclear surface layers are to be found farther out from the center than in the smooth undisturbed drop at zero temperature. The diagram of Fig. 37 shows the nuclear a Q n~\ OR OR OR (a) (b) (c) Fig. 37. Schematic plot of density of nuclear matter in a cross-sectional cut through a nucleus. In (a) is shown the abstract model of a uniform sphere without surface effects; in (b), the more realistic version of the Fermi model, with a fringe of grad- ually falling nucleon density; in (c), the nuclear density in a [highly excited and hence much expanded and strongly oscillating nucleus. density distribution at zero excitation and at high temperature; there is both a lowering of the mean p and a wide fringe of nuclear matter as t increases. An estimate of a 2 , the mean square amplitude of the surface vibration normal modes at a given temperature, has been made by Bagge, who shows that although the number of modes excited is not large, and 182 A Survey of Nuclear Reactions [Pt. VI quantum treatment is necessary, the value of a 2 is nevertheless very closely proportional simply to the nuclear temperature. This can be described by saying that the surface vibration energy is simply a definite fraction of the total excitation (F14) and that the mean frequency of the modes excited changes slowly. The surface effect appears to be quite important, reducing the barrier in the ratio V'/V — 1/(1 + "^ a 2 /R), and a semi-empirical estimate of the effective proton barrier as V p ' = 4/(1 + 0.15r) seems to fit the present data reasonably well. With this variation of barrier with temperature, the spectrum of emitted charged particles of course changes, and this reduction of V is fairly surely observed (Bll, B8). 2. The excited nucleus has a different balance between protons and neutrons; at high excitation there is a tendency toward increasing the difference of neutron and proton number, N — Z. This of course tends to favor the emission of protons from a nucleus already rich in neutrons. The tendency originates from a complex set of causes, most important of which is a new balance between potential and kinetic energy which has to be struck for the now expanded volume of the heated nucleus. A shift in volume decreases the Fermi kinetic energy, and the potential energy will readjust to fix a new minimum total energy, now at a different value of v from that of the cold nucleus of the same A. The effect has been evaluated by LeCouteur most recently, using a Fermi gas plus interactions. He shows that it leads to an entropy term of the form + const [(N — Z)/A] 2 . It becomes appreciable only for excitations above about 300 Mev. It has only a rather small influence therefore. It more or less corrects the high-energy emission for the fact that stable heavy nuclei in the ground state are neutron-rich, and equalizes the proton-neutron emission at high excitation. 3. The surface tension energy at high temperature decreases. Evi- dently the nuclear expansion could reach a "critical point" where the nucleons are all excited beyond their binding energies; here the drop would vaporize entirely. The nuclear surface tension will decrease regularly with temperature until the critical temperature, t c , is reached. Using the rough Fermi-gas-like relation between excitation and tem- perature, we can write W c = 8A Mev = At c 2 /10, and from this we get t c = 9 Mev, independent of A in this approximation. Now let us again take thermodynamics as a guide (G7). If we write down the Gibbs free energy for a surface film, in which the surface tension —7, surface force per unit length in the film, takes the place of pressure in the usual case, we have Gs — U — tS — yA for an area of film A, with internal energy U and entropy S at temperature r. Now the partial derivative — 8Gs/dA\ T = y gives the surface tension. An isothermal change in Sec. 11D] Nuclear Reactions at High Energy 183 film area will obey the first and second laws: dU = rdS + y dA, and we can write dGs = —Sdr — A dy. Differentiating, — dy/dr = (S/A) + (dGs/dr). But, in equilibrium at any r, the Gibbs free energy of film and bulk liquid are equal for a given mass of nuclear matter. We write G s = G L . Then -dy/dr = (S/A) + (l/A)(dG L /dr) = (S - S L /A). Now we can form Gs/A = Gl/A = — y + (U/A) — (rS/A), and we get a differential expression for the surface tension 7 as a function of r : -T + dy _ (U- Ul) dr A (134) The term pV in the liquid-phase free energy is cancelled by that of the surface film, except for thickness changes, which we neglect as usual. Mev 9 Temperature (Mev) Fig. 38. The nuclear surface tension as a function of nuclear temperature. The curves indicate different models: curve (a) arises from the actual surface energy of a classical liquid drop with surface waves excited by the thermal motion; curve (b) from the simple linear assumption for the integration of the free energy equation of the text. The two are very close. From (Yl). Now the right-hand side of (134) is the additional energy due to the surface film, measured per unit area of nuclear surface. Using as unit of area just 4tt 2 , this becomes about 14 Mev for nuclei in their ground states, with r = 0, as the semi-empirical formula showed in Section 10. As the temperature rises, surface vibrations are excited and the energy increases. Using a liquid drop model, Bethe (B13) has shown that the excess surface energy increases like r A /y(r). As a first approximation we can neglect the change in y, and write U — Ul/A = 14 + const t a . We can then solve (134), using as boundary condition the requirement that y(t c ) = 0. The surface tension then goes to zero, following the law 7(7") = 14 + t/t c — 15(t/t c Y a . A numerical integration has been per- 184 A Survey of Nuclear Reactions [Pt. VI formed by Yamaguchi (Yl) which gives a very similar result. A plot is given in Fig. 38. The reduced surface tension means that the balance between Coulomb and surface energy is displaced in favor of the Coulomb repulsion. We would expect this, since the Coulomb forces show only a slow change with mean separation, while the surface forces are, of course, of very short range. An expanded nucleus requires little additional energy to undergo fission, for which process the repulsive Coulomb forces over- come the surface tension; the nuclear volume energy can be taken as remaining constant throughout. The familiar Bohr-Wheeler estimate of fission threshold, as given in Eq. (107), is just E f = iTr 2 A 2A -y(r)f Q With this model, Fujimoto and Yamaguchi (F12) have estimated that the silver fission threshold would fall from some 50 Mev for the ground state of the nucleus to half of that value at about 100-Mev excitation, and to only a few Mev at 350-Mev excitation. The fission width can be estimated in the usual way as about 1/ ~ (r/27r)e~ E//T . This makes such fission a good competitor for really large excitation, 300 to 400 Mev or more. Such heavy spallation fragments are found not infrequently, as we have mentioned above (H3) . Of course, fission yielding fragments with much smaller mass is even more favored by the decrease in surface tension, and many reasonably heavy fragments have been seen in high- energy stars, especially the easily recognized unstable ones like Li 8 . The mechanism for fission here described is only a single one which may contribute to fission at high excitation energies. With the heaviest nuclei, and at somewhat lower energies, fission is observed subsequent to the emission of many neutrons, at temperatures below the barrier height for protons. This upsets the Coulomb-surface energy balance as well, and reduces the fission threshold. The mechanism is plainly very different (G4). It has been observed (F14) that a calculation of the temperature dependence of the surface energy, using the model for surface vibrations which Bagge employed, yields not very different results, and still predicts very easy loss of highly charged fragments at high temperature. Using Bagge's formulas for the mean amplitude of surface waves, but taking into account the variation of surface tension with temperature, it appears that excitations of 300 or 400 Mev would greatly distort the surface shape of nuclei, so much indeed that whole little drops would come off in a kind of spray, (a 2 ) l/% /R ^ 1 describes such a condition. It seems Sec. 11 D] Nuclear Reactions at High Energy 185 rather likely that the actual situation is more nearly a non-equilibrium local heating, capable of distorting the surface locally enough to pinch off a small drop, but not involving the surface of the entire nucleus in such a way as to give the nearly flat distribution of fission frag- ment masses which the near-disappearance of the equilibrium surface tension might lead one to expect. The whole phenomenon is clearly of importance, but our present account is too closely classical to trust quantitatively. 4. Fluctuations in Cooling. We have so far described in detail only the average behavior of the cooling nucleus; we have ignored all fluc- tuations. Obviously, such a complicated process in which only a few, or perhaps a few dozen, particles take part must be subject to important fluctuations in the number, type, energy, and direction of emission of the emitted nucleons. So far this problem is only sketchily understood, but a simple model will serve us at least for orientation (F12). Consider a nucleus so very hot that we can ignore its cooling, even though it has emitted many nucleons. It is restricted, moreover, to the emission of only one type of particle, with binding energy B, which also remains constant throughout the process. Then the prob- ability of emitting a single nucleon in energy range dT is just P(T) dT — e — T l rr r dT/r 2 , where t is the temperature, taken as constant through- out, and T the kinetic energy. Now the probability that the nucleus will emit n or more nucleons, when it has an initial total excitation W, is given by the multiple integral: P(n,W) =J ■ -jdT, ■ ■ -dT n _ l P{T i )P{T 2 ) ■ ■ -PiT^) (135) n— 1 < Z Ti < W - nB l where the upper limit follows because there must remain at least B Mev of excitation after the loss of n — 1 nucleons if at least one more is to be emitted, and the lower limit represents the process occurring by the possible route in which each of the first n — 1 nucleons just manages to dribble out with zero kinetic energy. It will not seriously distort this rather unrealistic picture to replace the Maxwell distribution in kinetic energy by a Gaussian distribution adjusted to have a mean and a standard deviation equal to the Max- wellian values. We know that T = 2t and (T — T) 2 = 2t 2 , so that we can rewrite _ , - ( T - ~T) V*r 2 d\T — T) 2(tt) 3 P(T) dT = e -< r -*W \ t H ' (136) 186 A Survey of Nuclear Reactions [Pt. VI Now the probability for the multiple emission is a compounding of nor- mal distribution. As in the familiar statistical problem of the chi-square «-i _ test, we write x 2 = 53 (Ti — T) 2 /4t 2 and recall that x 2 is then the 1 square of the radius vector in a hyperspace of n — 1 dimensions. In such a space the volume element is proportional to the quantity x" -2 dx- The proportionality constant is of course a function of n. If we recall that, as the excitation W grows without limit P(n, W) — > 1 for any given value of n, we can evaluate this function of n. Now for not too small values of (W — nB)/r and of n, the region of integration, which is bounded by the coordinate axes and a set of planes in the hyperspace, can be replaced without serious error of a sphere of radius say Xm in the polar coordinate system described by x- Remembering the required normalization, we obtain /Xmax /*'max X n ~ 2 e~ x d x =f{n)\ t in -*>' a e-* dt (137) which we can write P(n,W) = y[(n - l)./2, * max (W)]/r[(ra - l)/2], where y(n,x) is the incomplete gamma-function defined by y(n,x) = j e~H n ~ l dt and Y{n) = y(n, #-»«>) It is important to note that we want the dependence of P(n,W) not on the value of W, as in the usual statistical problem, but on the parameter n. We can estimate the distribution in x 2 > for any fixed large value of n, by the familiar saddle-point method, getting a Gaussian distribution given by e^* 2 -^- 1 )] 2 /^-!) 2 dy ? n ow t jj e fi rs t approximation to P(n) can be obtained by considering the geometrical interpretation of the integral, and observing that P(n) is unity for any n such that the sharp peak in x 2 lies within the original region of integration. Taking the original limits, then, and using the hyperplane rather than the sphere boundary, we can estimate that x 2 (H r ,w) = (W — nB)/2r. In the next approximation, the width of the region in which P(ri) falls from unity to zero can be estimated from the standard deviation in x 2 - We are led to these results for the mean number of nucleons emitted in the process and for the standard deviation, with an approximately Gaussian distribution in n: W + 2r ll S 2r + B (138) -= „ 8t 2 W+2t n ■ — n ~ B 2 2r + B Nuclear Reactions at High Energy 187 Sec. 11D] The relative rms fluctuation is just An/n ~ 3/(n) M - (r/B). The mean energy loss for the emission of a single Maxwellian particle is B + It, and the many-particle case is seen to give a mean value not very different from what one would get neglecting any correlation, namely W/(2r + B). The standard deviation is also not much different from the consequence of a very naive statistical estimate. Some reactions have been observed in which a highly excited nucleus emits a considerable but definite a -a O «3 ■§ ° PL, 0.3 y — ^* *v W- . t . , = 100 Mev initial 0.2 _ / / / / / / X^V / / N. \ 0.1 1 2 4 6 Number of evaporated particles (w) Fig. 39. The probability of emission of exactly n evaporation particles as a function of n. These fluctuations are calculated, on a very much simplified model, for a nucleus of initial A = 100 and excitation energy of 100 Mev. The dashed curve comes from an attempt to integrate the coupled equations of evaporation; the solid curve, which is a Poisson distribution, from a still simpler theory. Only qualitative conclusions may be safely drawn from the present theory. From (Tl) and (H10). number of identical nucleons, and the energy spectrum for those emitted particles seems to resemble the statistical energy distribution P(n,W) given above. 1 The omission of all the complicating effects from this picture is all too clear. A couple of attempts to solve the cascade equations under more realistic assumptions have been made (H10, Tl). Even these more elaborate efforts have neglected the change in mass and in binding energy as the cooling proceeds, and have in general neglected the fluctuation in energy taken off with each nucleon. The resulting fluctuations primarily emphasize the choice the cooling nucleus makes between protons and neutrons, with differing mean probabilities and mean energy removal for the two types of nucleons emitted. The results indicate something very close to a Poisson distribution in the number of neutrons or in the number of protons emitted, with the mean being given by the mean cooling behavior we have already considered. In Fig. 39 we plot some consequences of the two methods mentioned. It is fairly clear that most 1 Private communication from E. Segre. 188 A Survey of Nuclear Reactions [Pt. VI of the details of the process have been suppressed in these generalized models. Whether the fluctuations are underestimated or overestimated by these models is not clear. The changes in mass number and energy which have been neglected certainly cause correlations to be more im- portant, but some of the correlations have the effect of stabilizing the process closer to the mean. It is very likely here also that a Monte Carlo procedure will be the first to yield a reliable answer to the problem. E. Mesons : Virtual and Real We have come to the end of a sufficiently long, but yet only very sum- mary, account of the theory of nuclear reactions in general. It is ap- propriate to close with an admonition. All our considerations have been based on a single picture of the nucleus: physically, as a collection of neutrons and protons with more or less strong interaction; formally, as a system whose Hamiltonian was a sum of kinetic energies of a given number of heavy nucleons plus more or less complicated interaction potential energies. It is clear that this idea, taken either physically or formally, is at best an incomplete model. Looking at the nuclear system in short time intervals, or with fine distance resolution, we would not in fact distinguish an unchanging number of nucleons colliding and shifting position, but fixed in number and type. On the contrary, we know that even the exchange of mo- mentum which is the ordinary force between nucleons, like the exchange of spin and of charge which correspond to more subtle but still phe- nomologically described interactions is not to be understood in this way. The mediation of some kind of field of mesons seems certain. Like all fields whose quanta need relativistic description, that of the mesons within the nucleus cannot be assigned a definite particle count. We have to think of transient states, in which mesons appear and disappear, carrying momentum, charge, and spin back and forth among the heavy nucleons in what may be a very complicated manner. In a way we have used a description of the nucleus whose molecular analogue would be a pair of protons and an oxygen nucleus held together by some given potential, to form a molecule of water. Yet the meson field is still more complicated than that of the atomic electrons, though it contains charges. It is in some ways more closely akin to the essentially relativistic electro- magnetic field, in the cases where the photons may have a purely tran- sient existence, as ordinary photons do in fact have within the near-zone field of a radiating dipole. The anomalous nucleon magnetic moments arise from such meson fields. The existence of many-body forces, of ve- locity-dependent forces, of circulating currents not ascribable simply to orbital motion or to spin nipping of the nucleons is the very least to be Sec. HE] Nuclear Reactions at High Energy 189 expected from the virtual mesons within the nucleus (SI). Suspected are severe changes in the familiar properties of free nucleons themselves due to their immersion in the virtual meson sea (M8) . Transient changes in charge — producing doubly charged protons or negative neutrons — changes in spin, even in rest mass, are all probable circumstances of the nuclear matter, looked at sharply enough. And, when energy enough becomes available from outside to satisfy the demands of permanent meson liberation, all sorts of new effects can be seen. Meson absorption, emission, and scattering, where the mesons fly free of the nuclear region to reach our counters or the emulsion grains, are but the most obvious effects. All have been studied, and are in active development. What of the release of mesons, strongly interacting with nucleons as they do, which never leave the nucleus, but are reabsorbed within it to transfer energy, momentum, spin, and charge throughout the nuclear drop? Al- ready it seems that the sharp rise of the deuteron photo-cross-section (B6) in the energy region near and just above the threshold for free meson production involves some such explanations. All this is far from our simple picture of a closed and tight cluster of Z protons and N neutrons. But it is a picture closer to the truth. It is well to end this account, then, with the clear warning that application of the simpler ideas is reliable at best only in the domain where such long time averages are involved that the transient mesons can usually be replaced by the smooth forces of our picture, and for energies well below the energy at which real mesons can be released, even internally. With that warning not forgotten, the theory of nuclear reactions can yield results valuable for innumerable applications, for the detailed understanding of nuclear structure in a kind of chemist's approximation. 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Complete definitions and theory are found in text references (C5, K7, Y5). 5.0 ffiff0.15fm.t_ 4.0 Up lpjflfjg 'Tl'(?)'i$$$f 3 HC I 06||$1 "V ( rr} 2.0 IHo'B ^Po°7| BoM M0.9lllHlllf 3 1.0 0.2 0.4 X=E/E barri( . r 0.6 0.8 Fig. A-l. Function for computing the penetrability of the Coulomb barrier without angular momentum, Po, for 1 = 0. The penetrability Po is defined as: Pq = e~ 2c ° with Co = <7y(x). y(x) is here plotted. Figures 12d and 13a of the text give barrier heights for various nuclei, g is plotted in Figs. A-2a and A-2b. After (B13). 194 A Survey of Nuclear Reactions [Pt. VI 03 .a -a 03 Pt. VI] Appendix I 195 03 a, "3 J3 Ml 196 A Survey of Nuclear Reactions [Pt. VI to I 1.0 1 1 X=Vh Vi V2 < 1 1.0 1.5 o.c s = 2.0j 1.0 y=l(l + l)/g* 2.0 3.0 Fig. A-3. The -2ClXx.v) penetrabilities for angular momenta I ^ are defined as Pi = e _zo " x ' ! ". The plot gives (Ci — Co)/?, where g is the characteristic orbital momentum plotted in Figs. A-2a and A-2b as a function of two parameters: energy in terms of the barrier height x and parameter y = 1(1 + l)/<7 2 . After (K7). Ok 20 & ** s*% & X "1 € 15 11 20 30 40 50 60 70 80 90 Z X sA-U t w c\e 6 rvW^ »; V, , F '~*2£i •pvojjii-" Pr ot< )I\E *=0.15_ .20 30 40 50 60 70 80 90 Kg. A-4. Values of orbital angular momentum l c , such that the contribution to the total cross section for compound nucleus formation, a = ttX 2 2 (21 + l)Pi, 1-0 for all I greater than l c , is less than 10 percent. These curves form a guide for stopping the computation of the Pi as I grows. 197 198 z (atomic number) 100- 90- 80- 70- 60- 50- 30— 25 20— A Survey of Nuclear Reactions [Pt. VI 10— 9 8 7 6— 5 (a) E in Mev (center of. mass) 30- 20— 15- 10- 9- 8- 7 6- 5- 2 1.5- 1.0- .9- .8- .7- .6- .5- .4— .1— X V and d 1.0- .7- .6- .5- -1.0 — .7 — .6 (a') .1— .09- - .2 (a") 10" 10" '2 10" ■I io-« 5 2 10" s 4 6 8 io- 2 2 4 6 8 .i .2 .3 Hb) .5 .6 - .7 - .9 - .95 a p .2 .09— .1 .15- .2— .8- — .4 .09 .10 -.15 -.2 - .7 (c) -.95 Fig. A-o. Nomogram constructed for the evaluation of the penetrability of the Coulomb barrier, in the case of zero orbital angular momentum, I = 0. The penetrability Po can be found in two ways, of different accuracy: (1) Using lines (a), (a'), and (a"), the value of the parameter x, which is the energy of the particle measured in terms of the barrier height for a definite Z, can be found directly. The approximation A = 2Z is used for this graph, which is quite rough for the heavy elements, and may lead to Po in error by a factor of ~S, even where Po ~ 0.1. With this value of x, connecting points on lines (a), (6), and (c) then gives the penetrability, P . (2) The barrier height B may be found, not from the nomogram, but by directly reading off its value from the graphs of Figs. 12c and 13c of the text. Then x = E/E, can be found, and connecting proper points on (a), (6), and (c) of the nomogram will give Po to good accuracy over the whole readable range of the nomogram. Pt. VI] Appendix II 199 APPENDIX II We list first a number of theoretical and experimental survey papers of both a general and a more specialized kind. Compilations of Experimental Information M. S. Livingston and H. A. Bethe, Revs.. Modern Phys., 9, 245 (1937). List and discussion of all work on reactions up to July, 1937. J. Mattauch and S. Fliigge, Nuclear Physics Tables, New York, Interscience, 1946. Some data and references for all reactions up to about 1941. W. Hornyak, T. Lauritsen, P. Morrison, and W. A. Fowler, Revs. Modern Phys., 22, 291 (1951). A very full survey up to the middle of 1950 dealing with reac- tions involving the lightest target nuclei only, from H to Ne. Books S. Devons, The Excited Slates of Nuclei, Cambridge University Press, 1949. Excel- lent account of experimental methods and of theory. V. F. Weisskopf, in Lecture Series in Nuclear Physics, MDDC-1175, U. S. Govern- ment Printing Office, 1947, pp. 106 et seq. Theory. J. Blatt and V. Weisskopf, Theoretical Nuclear Physics, John Wiley & Sons, New York, 1952. Comprehensive account. Papers of More Restricted Scope (a) Photonuclear Reactions Experimental K. Strauch, Phys. Rev., 81, 973 (1951). Work with ~300-Mev y's. A. K. Mann and J. Halpern, Phys. Rev., 82, 733 (1951). L. Katz and A. S. Penfold, Phys. Rev., 81, 815 (1951). Theoretical J. S. Levinger and H. A. Bethe, Phys. Rev., 78, 115 (1950). E. P. Courant, Phys. Rev., 82, 703 (1951). (b) Theory of Particle Reactions in General V. F. Weisskopf, Helv. Phys. Acta, 23, 187 (1950). Physical picture. E. P. Wigner, Am. J. Phys., 17, 99 (1949). Introduction to the general dis- persion theory. E. P. Wigner and L. Eisenbud, Phys. Rev., 72, 29 (1947). Presents the full theory. J. Jackson, Phys. Rev., 83, 301 (1951). A simplification of the theory. T. Teichmann, Phys. Rev., 77, 506 (1950). E. Wigner, Phys. Rev., 73, 1002 (1948). Thelast two present instructive and useful applications of Wigner s gen- eral theory. (c) Neutron Excitation Theoretical E. Fermi and L. Marshall, Phys. Rev., 71, 666 (1947). Thermal neu- trons; includes experimental work. B. T. Feld, Phys. Rev., 76, 1115 (1949). Fast neutrons. H. Feshbach, D. Peaslee, and V. Weisskopf, Phys. Rev., 71, 145, 564 (1947). The basis for our version of the theory of dispersion. 200 A Survey of Nuclear Reactions [Pt. VI Experimental R. K. Adair, Revs. Modern Phys., 22, 249 (1950). A comprehensive review. D. J. Hughes, W. Spatz, and N. Goldstein, Phys. Rev., 75, 1781 (1949). (d) Charged Particle Excitation Theoretical V. Weisskopf and D. Ewing, Phys. Rev., 57, 472, 935 (1940). D. Peaslee, Phys. Rev., 74, 1001 (1948). Deuterons only. Experimental E. L. Kelly, thesis, University of California, Berkeley, UCRL-1044, 1951. S. N. Ghoshal, Phys. Rev., 80, 939 (1950). P. C. Gugelot, Phys. Rev., 81, 51 (1951). (e) Nuclear Fission Theoretical N. Bohr and J. Wheeler, Phys. Rev., 56, 426 (1939). General theory, using drop model. D. Brunton, Phys. Rev., 76, 1798 (1949). Experimental Plutonium Project, Revs. Modern Phys., 18, 513 (1946). J. Jungerman and S. C. Wright, Phys. Rev., 76, 111 (1949). J. Jungerman, Phys. Rev., 79, 632 (1950). Alpha- and alpha-induced fission. (f) High-Energy Nuclear Stars Theoretical W. Horning and L. Baumhoff, Phys. Rev., 75, 370 (1949). M. L. Goldberger, Phys. Rev., 74, 1269 (1948). Y. Fujimoto and Y. Yamaguchi, Prog. Theor. Phys., 4, 468 (1950). Experimental G. Bernardini, E. Booth, and S. Lindenbaum, Phys. Rev., 80, 905 (1950). 400-Mev protons. R. Menon, H. Muirhead, and O. Rochat, Phil. Mag., 41, 583 (1950). Pi- meson-induced' stars. Papebs on Specific Reactions The reactions are grouped by their means of excitation (list closed July, 1951). The references for each reaction are coded according to the first two or three letters in their authors' names and the years the works were published. A star (*) means that work above 100 Mev is reported. Some conventions are worth pointing out: 1. Strictly speaking, gamma-rays often are secondary products of other reactions. These are not separately considered. Only reactions in which gammas are the sole product are listed under "Gamma-rays." 2. Electron-induced reactions in which the electron is in fact captured by inverse beta-decay have not yet been observed. The electron serves to excite the nucleus, and itself passes on. It comes away from the reaction, though it is not in fact a reaction product. 3. Reactions with more than two product particles are listed only once; the order of emission (if known) is not taken into account. The classification here adopted depends upon the facts: (i) for evident reasons of effectiveness and availability, the incident particles used in almost all nuclear reac- Pt. VI] Appendix II 201 tions are light nuclei, up to helium (plus photons and electrons, etc.); and (ii) most reactions yield for at least one of the products another light nuclear particle, leaving behind as the second product particle a residual nucleus not far removed from the target nucleus in charge or mass number. Gamma-rays ( T , T ) Sc46; Wi45b; Wi45c; Gu41; Gae49. ( T , meson) *Lax51; *Lit51; *Mo50; *Pet51; *Ste50. ( T , n) Bal46; *Per49; Mc49; By51; Jo50; Kat51b; Med50; Pri50. (7, 2«) Bal46; *Per48. ( T , p) Same as (7, n) Hi47; Cou51; Di50; *Lev51; To51; *Wak51; By51. ( T ; p, n, or d) Bal46; *Per48; By51. (7, a) Has51; Mil50; Pr50. (7, multiple products) Bal46; *Per48. (7, star) *Gae50. (7, fission) Ti49; Hax41; Bo39; Ko50; *Su50. Electrons (e; e', n) B149; Sk48. (e; e') Wi44; Wi45a; Wi45c; Mul51. Mesons O, it) *Ca51; *Be50. (*-, star) *Be50; *Che50; *Men50; *Ta50. Neutrons (n, 7) Fes47; *Kn49; Hug49; Gos47; An50; Cap51; Ham50; He50; Hu51; Mu50; Ki51. (n, n) Fe47; Fel49; Se49; Fes47; Ad49; Har50; *Pas50; Stt51. (n, In) Hou46; Ma42; Hey37; Coh51; Fow50; Waf50. (n, spallation) De48. (n, p) Coo49; Boo37; Am35; Sa40; Coh51. (n; p, n, or d) *Kn49; *Bru49; *Chw50; *Ha50; Waf50. (n; H 3 or He 3 ) Bru49; Cor41; On40; *Ha50. (n, a) Sh41; Wu40; *Kn49; Am35; Sa40; B046; St50. (n, stars) *Go48; *Tr50. (n, fission) P146; *Ke48; Ph49; Fra47; Bou50; Ros50. Protons (p, 7) Du38; Wal48; Ben46; Fo48; Dev49; Hal50. (p, t) *Bj50; *Blo51; *Hen51; *Jon50. (p, n) Wei40; Du38; Del39; Bla51; *Bod51; *Kn51; *Mir51. (p, In) Te47a. (p, spallation) *Baz50; *Baz51; *Hy50; *Me51. (p, p) Fu48; Ba39; Hei47; Bed49; Goh51; Lth50; Rh50. (p; p, n, or d) Ri46; Pa48; Th49. (p, a) Th49; Bur49; Bar50; Ch50; Coc49; Ra50; Dev49. (p, heavy particle) *Mar51; *Wr50. (p, star) *Tho49; *Cam50; *Fr50; *Hod51; *Pek50. (p, fission) Ju48; Bo39. Deuterons (d, 7) We43. (d, n) Pe48; Ke49; C146b; Ro47; Amm49; Fa49; *Kn51. (d, 2ra) Ke49; Te47a. 202 A Survey of Nuclear Reactions [Pt. VI (d, spallation) Wil48; *Mi48; *Bat50; *Lin50. (d, p) Pe48; Pol49; C146b; A1148; Buw50; Cu50; Hav51; Phi50; Va51. (d, pa) C147. (d, d) Gug47; Gr49; Ker51. (d, H 3 ) Kr41; Wi46; Ka49. (d, a ) Kri49; Li38; C146a; In50; Sch50. (d, an) Mad50. {d, stars) *Ga49a; *Ho49. (d, fission) *Wo49; Ju48; Kr40; *Goe49. Tritium (H 3 , n) Cr51. (H 3 , p) Ku48a. (H 3 , He 3 ) Ku47b. He 3 (He 3 , p) A139. Alpha-particles (a, 7) Ben51. (a, mesons) *Bu49; *Jon51. (a, n) Eg48; Bra47; Rid37; Rie48; Hap49; Tem49. (a, 2n) Gh48; Te47a; Fi50; Tem49. (a, spallation) *Wo49; *Oc48; *Lin50. (a, p) Bro49; Roy51. (or, a) La39. (a; p, 3n) Hel46; Ne49; Te47b. (a, stars) *Ga49b. (a, fission) *Oc48; *Wo49; Ju48. 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McDaniel, Phys. Rev., 74, 315 (1948). (We43) K. Weiner, M. L. Pool, and J. Kurbatov, Phys. Rev., 63, 67 (1943). (Wei40) V. Weisskopf and D. Ewing, Phys. Rev., 57, 472, 935 (1940). (Theory and experiment: p, n.) (Wi44) M. L. Wiedenbeck, Phys. Rev. (Wi45a) M. L. Wiedenbeck, Phys. Rev., (Wi45b) M. L. Wiedenbeck, Phys. Rev. ; (Wi45c) M. L. Wiedenbeck, Phys. Rev. (Wi46) M. L. Wiedenbeck, Phys. Rev., 70, 435 (L) (1946). (Wil48) G. Wilkinson and H. Hicks, Phys. Rev., 74, 1733 (L) (1948). (Table, rare earths) (Wo49) R. D. Wolfe and N. E. Ballou, Phys. Rev., 75, 527 (L) (1949). (Wu40) C. S. Wu, Phys. Rev., 58, 926 (L) (1940). (Wr50) S. C. Wright, Phys. Rev., 79, 838 (1950). , 66, 36 (A) (1944). , 67, 59 (A) (1945). , 68, 1 (1945). , 68, 237 (1945). PART VII The Neutron BERNARD T. FELD The Massachusetts Institute of Technology In 1948, when this work was begun, the status of the field of neutron physics was uncertain and rather anomalous. Important progress, which had been made during the war, was known to the great body of physicists only through a few "releases" and through the Smyth report. The main prewar references — the articles of Bethe, Bacher, and Liv- ingston, in Reviews of Modern Physics — were hopelessly out-of-date. They had been partially and inadequately replaced by hastily assembled and informally distributed mimeographed notes of a Los Alamos lecture series on nuclear physics (LA-24) and of a series of lectures by Fermi on neutron physics. By the time the first draft of this work was completed, in June of 1949, the situation was quite a different one. As a consequence of a wise and far-sighted policy on the part of the American, British, and Canadian atomic energy commissions, practically all the basic scientific data, which had been accumulated during the war, appeared in the open literature. As a result of the widespread renewal of interest in the prob- lems of neutron physics, the field has developed, and continues to de- velop, at a rate which has converted the task of compiler and author into an almost hopeless struggle against obsolescence and has expanded the dimensions of this work far beyond its original conception. The unequal struggle was, quite arbitrarily, concluded as of July 1, 1951. Since then, some changes have been made, mainly as a result of impor- tant new developments which have come to the attention of the author, mostly through publication in American journals. This work is primarily intended for the practicing nuclear physicist. It presupposes a knowledge of the fundamentals of nuclear physics as well as of its terminology. Some of the terms, such as "barn," 1 are of comparatively recent origin, but their use has become widespread. 1 Barn, a unit of cross section; origin: big as a . (1 barn = 10 -24 cm 2 .) This unit is said to have been invented by the nuclear physicists at Purdue Uni- versity, around 1941 or 1942, to describe nuclear cross sections which are relatively easy to measure. 208 Sec. 1A] Properties and Fundamental Interactions 209 Although an attempt has been made to present herein a complete summary of the major aspects of neutron physics, some important applications have, perforce, received inadequate treatment or been omitted completely. Thus, no attention has been paid to the role of neutrons in biophysics; nor has any space been devoted to the many interesting problems concerning the effects of neutron interactions on the macroscopic physical properties of matter. Also, and regretfully, omitted is a discussion of the fascinating problems of the origin of the elements, in which the properties of neutron interactions may have played a decisive role. [The interested reader is referred to an excellent summary of this field by Alpher and Herman, Revs. Mod. Phys., 22, 153 (1950).] It would be impossible to give a complete list of the many individuals to whom I am indebted for aid, encouragement, information, discussion, and criticism during the progress of this work. My colleagues at M.I.T., Brookhaven, and elsewhere, especially H. Feshbach, M. Goldhaber, G. Placzek, and V. F. Weisskopf , have been unusually generous in this respect. I owe a special debt to E. Amaldi, of the University of Rome, not only for congenial and informative discussions but for having gen- erously allowed me access to an unpublished work on neutron physics by G. C. Wick and himself. In memory of many pleasant hours, during which the outline of this article achieved shape and substance, I respectfully dedicate this work to my teacher and friend, the late H. H. Goldsmith. SECTION 1. PROPERTIES AND FUNDAMENTAL INTERACTIONS A. Discovery The discovery of the neutron is one of the most dramatic chapters in the history of modern physics. It started in Germany, in the last month of 1930, with the report by Bothe and Becker (B50) of a penetrating radiation resulting from the bombardment of certain light elements by polonium alpha-particles. They bombarded many substances, using a Geiger point-counter as a detector of the resulting radiation. Most of the elements investigated (Pb, Ag, Ca, N, C, O, Ne) yielded no detect- able radiation; two (Mg, Al) showed a slight effect. Lithium, boron, and fluorine gave appreciable amounts of radiation capable of affecting the counter, and beryllium yielded a comparatively tremendous amount. Bothe and Becker concluded that the radiation consisted of gamma- 210 The Neutron [Pt. VII rays, more penetrating than any that had been observed up to that time. 1 Curie and Joliot (C36, J10) immediately undertook a study of the properties of this penetrating radiation. They had available a much stronger polonium source (100 millicuries, as compared to the 3-7 available to Bothe and Becker) and were able to measure the absorp- tion of the radiation in lead. They observed, for the radiation from beryllium, an exponential attenuation with an absorption coefficient of 0.15 cm -1 . (The radiations from boron and lithium had lead absorp- tion coefficients of 0.2 and 1.7 cm -1 , respectively.) So far, there was nothing to contradict the suggestion — and, indeed, this was universally assumed — that the radiations were very penetrat- ing gamma-rays. (We know now that an absorption coefficient in lead of 0.15 cm -1 is smaller than that of the most penetrating gamma-rays.) However, in January of 1932, Curie and Joliot (C37) reported the fol- lowing interesting observations : They investigated the effects of placing thin screens of various materials in front of the ionization chamber, which was being used to detect the radiations. For most of the screens nothing noteworthy occurred. However, when the screens contained hydrogen, the current in the ionization chamber went up. Curie and Joliot inferred that the increased ionization was due to the ejection of protons from the screen by the primary radiation. They strengthened this theory by the following set of observations: (1) The application of a magnetic field in the region between the screen and the ionization chamber did not decrease the effect; it would have, if the ejected particles had been slow electrons. (2) The effect vanished when 0.2 mm of aluminum was placed between the screen and the chamber. This was sufficient to absorb protons, but not fast electrons. (3) Cloud chamber photographs of the tracks of particles ejected by the radiation 1 The conclusions of Bothe and Becker, as well as the observations upon which they were based, turn out, in retrospect, to be completely valid. The Geiger point- counter, used as a detector by these investigators, was not sensitive to neutrons; thus, Bothe and Becker could not have observed neutrons in their experiments. It was only with the introduction of ionization chambers, proportional counters, and cloud chambers as detectors that the neutrons became observable in subsequent investigations. The measurements of the gamma-ray energies (~3 Mev from boron and ~5 Mev from beryllium), reported by Bothe and Becker, have subsequently been confirmed. Indeed, the important discovery by Bothe and Becker of artificial excitation of nuclear gamma-radiation, reported in their 1930 paper, has unfortu- nately been almost completely obscured by the drama associated with the discovery of the neutron. For a more complete discussion of the significance of the observa- tions of Bothe and co-workers, the reader is referred to an article by Fleischmann (F32). Sec. 1A] Properties and Fundamental Interactions 211 from hydrogen-containing screens showed ionization consistent with that of protons, but inconsistent with that of electrons (C38). Curie and Joliot first hypothesized that the ejection of protons from the screens was due to Compton scattering of the incident gamma-radia- tion by the hydrogen nuclei. From the observed range (energy) of the (recoil) protons, they estimated the energy of the photons from beryl- lium to be 50 Mev. Curie and Joliot recognized a number of serious difficulties in connection with their hypothesis; they could conceive of no source of such high-energy photons in a reaction of alpha-particles on beryllium, and they felt (erroneously) that photons of such high energy should be even more penetrating than the observed radiation. But, while they decided that the ejection of protons (and other light nuclei) was by a new type of gamma-ray interaction (C38), it remained for Chadwick, working at the Cavendish Laboratory in England, to reject the gamma-ray hypothesis and take the bold step of postulating a new particle. Chadwick was very quick to follow up the researches reported from the Institut du Radium. Only slightly more than a month after the report discussed above he was publishing data (C5), obtained with counters and cloud chambers, showing that the radiations from beryl- lium bombarded with alpha-particles were capable of conferring high speeds not only upon protons, but also upon the nuclei of other light elements (He, Li, Be, B, C, N, 0, A). From the observed ranges of the light nuclei, and using the then current range-energy relationships, Chadwick showed that the Compton recoil hypothesis of Curie and Joliot was inconsistent with the data. The data could, however, be explained if the light nuclei were assumed to be recoils from elastic collisions with a neutral (to explain the great penetrability) particle of approximately protonic mass. To obtain the mass of this particle (called by Chadwick the neutron) Chadwick (C6) used the available data on the maximum range (velocity) of the proton recoils, and the results of Feather (F2) on the maximum range of the nitrogen recoils, observed in a cloud chamber. (If these were due to Compton recoils, they would have required gamma-ray energies of 55 and 90 Mev, respectively.) By application of the law of conservation of momentum, Chadwick derived that the particles re- sponsible for these recoils had a mass of 1.15 times the proton mass, with an uncertainty such that "it is legitimate to conclude that the mass of the neutron is very nearly the same as the mass of the proton." Another estimate of the mass of the neutron was made by Chadwick from observations on the neutron-producing reaction B 11 + He 4 — > N 14 + n 1 . On the assumption that the maximum-energy neutrons (obtained 212 The Neutron [Pt. VII from the maximum range of proton recoils) correspond to leaving N 14 in its ground state, and from the values of the masses of the three nuclei involved, previously determined by Aston, Chadwick deduced the value of 1.0067 atomic mass units for the mass of the neutron. Within a short time, immediately following upon the series of an- nouncements described above, a large number of investigators in many lands were conducting experiments on the properties of the neutron and its interactions. Although it is not the author's intention to continue this historical survey much beyond the discovery stage, it is of some interest to review briefly the advances made within only a year after the appearance of Chadwick's papers. For a more detailed discussion of the early history, the reader should refer to the original papers and to excellent review articles (written in 1933) by Chadwick (C7) and Darrow (D4). It was soon ascertained that the neutrons resulting from the bom- bardment of beryllium and boron by polonium alpha-particles had a rather wide energy spread (F2, M21, C39) including many neutrons of energy considerably below 1 Mev (A36), with the neutron energy strongly dependent on the direction of emission, with respect to the alpha-particle direction, as well as on the energy of the alpha-particle. Gamma-rays were also shown to be emitted in the neutron-producing reaction (B51, C39). The yield of neutrons from beryllium and boron was found to decrease rapidly with decreasing alpha-particle energy (R7, C40, C7). By placing considerable quantities of lead next to the neutron-detect- ing ionization chamber (but not directly in the beam) (B70) or by sur- rounding the cloud chamber with copper (A36), the number of neutrons detected was appreciably increased, thus indicating a large neutron scattering by these substances. In the nitrogen gas of the cloud chamber, used in the experiments of Feather (F2), a number of events were observed which could be as- cribed to the transmutation N 14 + n 1 — ► B 11 + He 4 , the reverse of the reaction on boron which had been observed to produce neutrons. He also observed transmutations in oxygen and carbon (F3). Transmu- tations were observed in nitrogen, oxygen, and aluminum by Meitner and Philipp (M21), and in nitrogen by Harkins, Gans, and Newson (H37) and by Kurie (K27). At first, it was generally assumed that the neutron is probably a closely bound combination of a proton and an electron, 1 especially since 1 As early as 1920, Rutherford (R27) published an interesting and prophetic specu- lation: "Under some conditions ... it may be possible for an electron to combine much more closely with the H nucleus [than in the neutral hydrogen atom], forming Sec. IB] Properties and Fundamental Interactions 213 the first estimates indicated that its mass is less than that of the proton. The first suggestion that the neutron should be regarded as a fundamen- tal particle appears to have come from Iwanenko (13), and it soon became clear, mainly on the basis of quantum-mechanical arguments involving the spin and statistics of light nuclei, that the neutron, like the proton, probably has a spin of J and obeys Fermi-Dirac statistics (M21, C7). Thus, with the discovery of the neutron and the recogni- tion of its properties, the currently accepted picture of nuclei, as con- sisting of protons and neutrons, soon emerged. Since 1933 the development of the field has proceeded with rapidly increasing intensity. We therefore abandon, at this point, the historical survey. Instead, we shall summarize and discuss the present knowledge of the properties of neutrons and their interactions. We shall, however, in discussing each aspect, attempt to include some of the historical back- ground. It must be emphasized that, although, in the relatively short time that has elapsed since the discovery of the neutron, tremendous progress has been made in understanding and utilizing it, there is still much to be done before the neutron can be said to be completely under- stood. Many important and crucial experiments are in the process of being performed, while others are still in the future. B. Properties Since the neutron is one of the constituents of atomic nuclei, a knowl- edge of its properties . is fundamental for the understanding of nuclei. In addition, the properties of the neutron determine, to a large extent, the interactions between neutrons and nuclei, and between neutrons and conglomerations of nuclei (matter). In this section we summarize these properties from the point of view of the neutron as a fundamental particle. Wherever possible, we shall discuss the experimental evidence on which the conclusions as to the nature of the neutron are based. How- ever, the understanding of many of these experiments depends on a detailed knowledge of the interaction of neutrons with matter. In such cases, the results of the experiments will simply be stated, and the dis- cussion of the experiments reserved for subsequent sections. a kind of neutral doublet. Such an atom would have very novel properties. Its external field would be practically zero, except very close to the nucleus, and in consequence it should be able to move freely through matter. Its presence would probably be difficult to detect by the spectroscope, and it may be impossible to contain it in a sealed vessel. On the other hand, it should enter readily the struc- ture of atoms, and may either unite with the nucleus or be disintegrated by its intense field, resulting possibly in the escape of a charged H atom or an electron or both." 214 The Neutron [Pt. VII Before proceeding with the discussion of the neutron, a few points should be noted in justification of devoting so much attention to the neutron as compared to other nuclear particles. The neutron is un- matched among atomic and nuclear particles as a tool for the investiga- tion of nuclear properties. The distinguishing feature of neutrons is the absence of electric charge; thus, the interaction of neutrons with matter is primarily determined by purely nuclear properties. In the fifty or so years since its beginnings, nuclear physics has made large contributions to the development of the other sciences (chemistry, geology, biology, medicine) and to technology. Since its discovery the neutron has assumed a most important role in furthering this progress. Were there no other reason, the importance of the neutron for the nu- clear chain reaction might constitute sufficient justification for assign- ing to it a position of special significance among the tools of modern science. 1. Charge. The neutron is usually assumed to have no net charge. This assumption is consistent with all the observed properties and inter- actions of neutrons. However, the observations do not preclude the possibility that the neutron may have a net charge so small as to have heretofore eluded detection. It is therefore of interest to derive from the available evidence an upper limit for the magnitude of the possible neutron charge. The most direct evidence on the neutron's neutrality comes from the experiments of Dee (D6), reported at the same time as Chadwick's announcement of the discovery of the neutron. He investigated the ionization produced in air in a cloud chamber irradiated by fast neu- trons and concluded that, if the neutron interacts with atomic electrons at all, this process produces not more than one ion pair per 3 meters of the neutron's path in air. From these data it may be concluded (F36) that the charge of the neutron is less than J^oo of the proton charge. A somewhat less direct determination of an upper limit to the neutron charge may be obtained from considerations involving the observed neu- trality of atoms throughout the periodic table. Since the ratio of neu- trons to protons in atomic nuclei increases from zero in hydrogen to 1.6 in uranium, this neutrality implies both that the difference between the proton and electron charges is small and that the neutron charge is small. An estimate of the, possible magnitude of the neutron charge depends on the accuracy with which we know atoms to be neutral. Rabi J and co-workers have observed that the molecule Csl (108 pro- tons and electrons and 152 neutrons) has a net charge less than 10 — 10 of 1 Private communication. Sec. IB] Properties and Fundamental Interactions 215 the electron charge. From this observation it is possible to draw a num- ber of conclusions: (1) The smallness of the charge of Csl may be due to an accidental cancellation of the neutron charge and the proton-electron charge difference. In this case, which is exceedingly unlikely, the mag- nitude of these charges can only be determined by another observation on some other atom or molecule, with a different neutron-proton ratio. In any event, both the neutron charge and the proton-electron charge difference would have to be quite small to account for the neutrality of atoms, say < 10 — 5 e. (2) Either the neutron charge or the proton-elec- tron charge difference is zero (or considerably less than 10~~ 12 electron charges), in which case the non-zero charge is less than 10 — 12 electron charges. (3) The neutron charge and the proton-electron charge dif- ference are both finite, and very small. Thus, if they were equal and opposite, the neutron charge would have to be less than 2 X 10 — 12 electron charges. It seems quite reasonable to conclude, from this evi- dence, that the net charge on the neutron is exceedingly small, probably less than 10 — 12 electron charges. Despite the electrically neutral character of the neutron, there is a very small electromagnetic interaction between neutrons and charged particles arising out of the magnetic dipole moment of the neutron (to be discussed). In addition, it should be pointed out that, according to currently prevailing theories, the neutron should not strictly be regarded as a fundamental particle, but rather as having a complex structure involving equal numbers of + and — charged particles (mesons). Such a structure is certainly required for the understanding of the mag- netic moment of the neutron. The complex nature of the neutron would lead to a small, purely electrical interaction between neutrons and elec- trons; the existence and magnitude of such a neutron-electron interac- tion is being investigated and is discussed further on. Two other points are worth noting as having a bearing on the possible charge of the neutron: (1) The observation of the decay of the neutron into a proton, an electron, and a neutrino (discussed in Section 1B3) implies that (a), if the proton and electron charges are equal and oppo- site, the neutron and the neutrino have the same charge, if any; or (b), if the neutrino is uncharged, the neutron charge is equal to the proton- electron charge difference; or (c) some combination of (a) and (b). In this connection, de Broglie has pointed out (B69) that hypothesis (a) would be consistent with the neutrality of atoms if there were both posi- tively and negatively charged neutrinos and, correspondingly, neutrons, so that heavy nuclei could contain approximately equal numbers of neu- trons of the two charges. (2) The smallness of the neutron-electron 216 The Neutron [Pt. VII interaction (Section 1C3) implies that the charge on the neutron must be exceedingly small. In particular, the observation by Fermi and Mar- shall (F23) that the scattering of thermal neutrons by xenon atoms is spherically symmetrical to better than 1 percent allows us to set an upper limit on the possible net charge of the neutron of >~10~ 18 electron charges. 2. Mass. 1 The most accurate determination of the neutron mass is obtained, indirectly, from observations on nuclear transmutations in which all the masses and energies, except the mass of the neutron, are known. Thus, the estimate of Chadwick, quoted above, was obtained from consideration of the B u (a,n)N u reaction (C7). In the same paper Chadwick observed that a more direct determination of the neu- tron mass could be obtained if the binding energy of the deuteron, the nucleus consisting of a neutron and a proton, were known, since the masses of the proton and deuteron had both been determined by mass- spectroscopic means. Soon afterward Chadwick and Goldhaber (C8) observed that the deuteron can be decomposed into a neutron and a proton by the absorp- tion of a gamma-ray from ThC" (2.614 8 Mev). From the energy of the resulting proton they obtained a value of 2. 1 Mev for the deuteron bind- ing energy. Until recently the accepted neutron mass was obtained from accurate measurements of the threshold gamma-ray energy for the photodisintegration of the deuteron. However, quite recently Bell and Elliott (B16) have accurately measured the energy of the gamma-rays resulting from the capture of neutrons, of negligible kinetic energy, by protons (the reverse of the photodisintegration). Using their value of 2.230 ± 0.007 Mev for the gamma-ray energy, and other data, Bainbridge (Volume I, Part 5) gives the value M = 1.008982 atomic mass units for the neutron mass. 3. Instability. With the first relatively accurate determination of the neutron mass (C8), which showed that it exceeds the mass of the hydro- gen atom, Chadwick and Goldhaber ventured the prediction that the neutron should be unstable against beta-decay, according to the reaction neutron — > proton + /3~ + neutrino. The maximum beta-ray kinetic energy for the decay of neutrons of negligible kinetic energy is given by the n — H mass difference, which we take to be equivalent to 782 ± 1 1 That the neutron mass behaves in the conventional fashion with respect to gravi- tation has been directly verified by McReynolds (M19), who measured the "free fall" of slow neutrons. Sec. IB] Properties and Fundamental Interactions 217 kev (T16). 1 Assuming that the decay of the neutron follows the empiri- cal laws of beta-decay, being an allowed transition with a log ft value of ~3.5, the half-life of the neutron should be ~20 min. Recent experiments in two laboratories have established that the neutron does decay, as per the above predictions. In the first ones, per- formed at the Oak Ridge National Laboratories by Snell, Miller, Pleason- ton, and McCord (S43, S44), the neutron decay has been observed through the simultaneous (coincidence) detection of the beta-particle 200 300 400 BOO Energy (kev) Fig. 1. Fermi plot of the negatrons from the decay of neutrons, due to Robson (R15). The deviations from the allowed shape, below 300 kev, are instrumental. and proton, resulting from the decay of neutrons in an intense beam passing through an evacuated chamber containing the particle detec- tors. Control experiments were performed to establish that the coinci- dences occurred only in the presence of the neutron beam, and that the beta-particles were of roughly the expected energy. These experiments are consistent with "a half-life in the range 10-30 minutes." Robson (R15) at the Chalk River Atomic Energy Project, in Canada, has not only observed the neutron decay and measured its half-life but, in an experiment of exemplary care and ingenuity, has also succeeded in measuring the spectrum of the decay electrons. He first observed the positive heavy particles resulting from the decays (in a highly sensitive mass spectrometer) and established that they are protons. He then 1 Actually, in the decay of a neutron at rest the emission of a ^-particle and a neutrino requires, for the conservation of momentum, that some energy be given to the proton. However, for the emission of a /3-ray of the maximum possible energy, the kinetic energy of the recoil proton is only 0.43 kev. 218 The Neutron [Pt. VII succeeded in obtaining coincidences between the protons and their associated beta-particles, detected in a magnetic lens electron spectrom- eter in which he simultaneously measured the energy of the beta-parti- cles. The beta-spectrum obtained in this manner is shown in Fig. ] , a conventional allowed-transition Fermi plot. The end point corresponds to a maximum beta-ray kinetic energy of 782 ± 13 kev. From an (ex- perimental plus computational) evaluation of the geometrical efficiency of proton detection and a measurement of the neutron density in the beam (only ~10 4 neutrons/cm 3 ), Robson deduced the half-life of 12.8 ±2.5 min for the decay of the neutron. Other experiments have been suggested for the observation of the decay of the neutron and the measurement of its half-life. Among the most interesting is the following: A sealed, evacuated, thin- walled con- tainer of material with low neutron absorption, say glass, is placed in a region of high neutron density. After some time has elapsed, the accumulation of hydrogen gas in the container is measured. The con- tainer acts as a semi-permeable membrane, permitting the free entry of neutrons but preventing the escape of hydrogen formed by neutron decay in the container. For a neutron density of about 0.5 X 10 10 neutrons/cm 3 (corresponding to a thermal neutron flux of ~10 15 neutrons • cm" -2 • sec -1 , of average velocity ~2 X 10 s cm/sec) a hydrogen pressure of ~10~ 4 mm of Hg would be developed in the container in approximately one month ; such an accumulation of hydrogen gas should be easily observable. The difficulties of this experiment involve the attainment of sufficiently high neutron fluxes (see Section 3), possible production of hydrogen through (w,p) reactions in the walls of the con- tainer, and the necessity for complete removal of hydrogen from the container (walls) prior to irradiation. 4. Spin and Statistics. Like the electron and the proton, the neutron has a spin of \ and obeys Fermi-Dirac statistics. The evidence for the spin value of \ is quite conclusive, although it involves a combination of experimental observations and the theoretical deductions therefrom. Some of these experiments involve the interaction between neutrons and protons and the properties of the deuteron; they will be further dis- cussed in Section 1C. The most important evidences for the value of \ for the spin of the neutron are: (1) the cross section and energy dependence of the scatter- ing of neutrons by protons, taken in conjunction with the evidence con- cerning the neutron-proton force, derived from the binding energy and spin of the deuteron; (2) the values of the magnetic moments of the Sec. IB] Properties and Fundamental Interactions 219 deuteron, proton, and neutron; (3) the coherent scattering of neutrons by hydrogen, as evidenced in experiments involving the scattering by hydrogen-containing crystals, total reflection from hydrogen containing "mirrors," and the scattering by ortho- and para-hydrogen; (4) the polarization of neutrons by scattering in ferromagnetic materials and, in particular, by total reflection from magnetic mirrors. The results of these experiments, coupled with the success of the semi-empirical theory of nuclear forces in explaining the observed properties of the deuteron and many of the properties of light nuclei, constitute a most convincing argument for the correctness of the assignment to the neutron of a spin of J. As a particle of odd half-integral spin, the neutron is expected to obey Fermi-Dirac statistics. All the available evidence supports this expectation. The most important arguments concerning the statistics obeyed by the neutron involve (1) properties of the light nuclei — in particular, the fact that the deuteron and alpha-particle are known to obey Bose-Einstein statistics — which can be qualitatively understood only by use of the exclusion principle (in a fashion closely analogous to its application to the qualitative understanding of the periodic table), and (2) the saturation of nuclear forces, as manifested in the constant density of nuclear matter (the proportionality of the nuclear radius to A*). 5. Magnetic Moment. 1 The first strong indication that the neutron has an intrinsic magnetic dipole moment came from the observation that the moments of the proton and deuteron are very different : nd < n P by <~2 nuclear magnetons. Since the deuteron is known to have spin I = 1, it is expected, on the basis of the simplest reasonable deuteron structure (a neutron and a proton in a 3 Si state), that m = y. p + \x n . 1 The neutron, having spin \, cannot show any moments higher than a dipole moment. Although the possibility of the existence of an electric dipole moment of the neutron is usually dismissed, on the basis of theoretical arguments involving the concepts of symmetry and parity, it has recently been pointed out by Purcell and Ramsey (P32) that the assumptions on which these arguments are based are still not completely proved. Thus, for example, "if the nucleon should spend part of its time asymmetrically dissociated into opposite magnetic poles of the type that Dirac has shown to be theoretically possible, a circulation of these magnetic poles could give rise to an electric dipole moment." Although the detection of a possible electric dipole moment of the neutron is experimentally rather difficult, the above authors suggested that it could be observed in a modification of the experiment of Alvarez and Bloch (A13); they proposed to detect a shift of the neutron precession frequency, caused by the application of a strong electric field. However, such an experiment by Purcell, Ramsey, and Smith (S40) yielded a negative result and placed an upper limit, on the neutron's electric dipole moment, of two opposite electron charges sepa- rated by a distance of 5-10 -21 cm. 220 The Neutron [Pt. VII The magnetic moment of the neutron has been measured by an in- genious modification of the Rabi-type molecular beam magnetic reso- nance experiment (Rl). In this experiment the usual polarizing and analyzing (focusing) inhomogeneous magnetic fields are replaced by slabs of magnetized iron whose property of preferentially transmitting neutrons of one direction of polarization is used in a manner completely analogous to an optical polarimeter. (The polarization of neutrons by transmission through magnetized iron, the Bloch effect, will be dis- cussed in Section 5D.) Early measurements of the neutron magnetic moment (F47, P28) gave a value of <~ — 2 nuclear magnetons. The negative sign, which was definitely established by observing the direction of precession of neutrons in a known magnetic field, means that the neutron angular momentum and its magnetic moment are oppositely directed. The first accurate measurement of the neutron moment, by Alvarez and Bloch (A13), yielded the value | n n \ = 1.935 ± 0.02 nuclear magne- tons. More recent experiments (A34, B39, R16) have measured, with high precision, the ratio of the neutron moment to the proton moment. These measurements, when combined with the latest value of the pro- ton moment, yield (Ml) fi n = -1.91280 ± 0.00009 nuclear magnetons 6. Wave Properties. In accordance with the laws of quantum me- chanics we expect that the neutron should, under the appropriate experi- mental conditions, exhibit wave properties. Associated with neutrons of kinetic energy E (velocity v, momentum p) there is a wavelength x = 1 = Wev> (for " <<c) (1) Table 1 gives the neutron wavelengths corresponding to an assortment of neutron energies. The energy is expressed both in electron volts and in degrees Kelvin, the temperature T corresponding to a given energy E being defined by the relationship E = kT (2) where E is expressed in ergs and k is Boltzmann's constant, k = 1.3803 X 10 -16 erg/°K. Also included in Table 1 is the "Dirac wavelength," X = X/2t, which is frequently the form in which the neutron wave- length enters into the expressions with which we shall be concerned. The classification of neutrons into types, according to their energy, will be described in Section 1B7. Sec. IB] Properties and Fundamental Interactions TABLE 1 Wavelengths Associated with Various Neutron Energies 221 E(ev) T(°K) v (cm/sec) X (cm) X (cm) Type 0.001 11.6 4.37 X 10 4 9.04 X 10" 8 1.44 X 10" 8 Cold 0.025 290 2.19 X 10 5 1.81 X 10" 8 2.88 X 10" 9 Thermal 1.0 1.16 X 10 4 1.38 X 10 6 2.86 X 10 ~ 9 4.55 X 10- 10 Slow (reso- nance) 100 1.16 X 10 6 1.38 X 10 7 2.86 X 10" 10 4.55 X 10- 11 Slow 10 4 1.16 X 10 8 1.38 X 10 8 2.86 X 10" 11 4.55 X 10" 12 Intermediate 10 6 1.16 X 10 10 1.38 X 10 9 2.86 X 10- 12 4.55 X 10 " 13 Fast 10 8 1.16 X 10 12 1.28 X 10 10 2.79 X 10~ 13 4.43 X 10- 14 Ultrafast 10 10 1.16 X 10 14 2.99 X 10 10 1.14 X 10- 14 1.81 X 10~ 16 Ultrafast (relativis- tic) For the purpose of rapid calculation it is convenient to note the fol- lowing relationships: r(in °K) = 1.16 X 10 4 # Kin cm/sec) = 1.38 X 10 G E y ° X(in cm) = 2.86 X lO -9 ^ K X(in cm) = 4.55 X 10 -10 £ -H where E is always given in electron volts: 1 ev = 1.602 X 10" ergs These expressions hold only in the non-relativistic energy region, i.e., for kinetic energies well below the energy corresponding to the neutron rest mass, Mc 2 = 939.5 million electron volts (Mev) As can be seen from Table 1, the relativistic effects (on the connection between v, or X, and E) are already evident, albeit still small, at a kinetic energy of 100 Mev. In general, the importance of the wave characteristics of the neutron is determined by the magnitude of the ratio between the neutron (Dirac) wavelength and the dimensions of the system with which the neutron is interacting. For neutrons of wavelength large compared to nuclear 222 The Neutron [Pt. VII dimensions, the wave properties are of primary importance in deter- mining the nature of the interaction between neutrons and nuclei. Since nuclear radii fall in the range 2 X 10~ 13 to 10 -12 cm, the wave properties of the neutron are seen to be important for energies up to the fast neutron region; for fast neutrons, the wave properties are of comparatively lesser significance. The wavelengths of thermal neutrons are of the same order as the interatomic distances in solid matter. Thus, we would expect neutrons of these energies to show interference effects in their passage through, and scattering by, ordered materials. Such interference phenomena were predicted by Elsasser (E5) and by Wick (W24) in 1936-37, and soon indicated by Preiswerk and von Halban (P29) and by Mitchell and Powers (M33). Indeed, thermal neutrons have been found to behave very much like x-rays under similar circumstances, with the important difference that the scattering and absorption of neutrons are nuclear phenomena, whereas the corresponding properties of x-rays arise from their interaction with atomic electrons. Neutron diffraction and inter- ference phenomena will be discussed in detail in Section 5. It is worth observing that, although wave effects are relatively less important for fast neutrons, they may possibly become significant again for ultrafast neutrons. In this energy range the neutron wavelengths are of the order of the distance between nucleons in the nucleus, and the scattering from the nucleons within a nucleus can exhibit interfer- ence effects. Indeed, it is possible that, with the development of tech- niques for studying such effects, investigations of the "form factor" in nuclear scattering may provide important information on nuclear structure. 7. Classification according to Energy. In the subsequent discussion we shall find it convenient to refer to neutrons of different kinetic ener- gies according to the following system of classification: I. Slow neutrons: < E < 1000 ev II. Intermediate neutrons : 1 kev < E < 500 kev III. Fast neutrons: 0.5 Mev < E < 10 Mev IV. Very fast neutrons: 10 Mev < E < 50 Mev V. Ultrafast neutrons: 50 Mev < E Although the dividing lines between these categories are quite arbi- trary, this system of classification can be justified on two grounds. In the first place, the interactions of the different classes of neutrons with Sec. IB] Properties and Fundamental Interactions 223 nuclei and with matter in bulk involve, in general, different reactions and types of phenomena. Secondly, the methods of producing and de- tecting the different classes of neutrons are quite different. (a) Slow Neutrons. The behavior of neutrons in this energy range has been more extensively investigated than in any of the other ranges. The interaction of slow neutrons with heavy nuclei is characterized by sharp absorption resonances and large absorption cross sections for very low-energy neutrons. This is also the region in which crystal effects are important. Of particular importance to the study of the slow neutron region is the fact that a number of instruments, known as monochromators, have been developed whereby neutrons of a given energy may be singled out and their properties studied. Monochromators in the slow neutron re- gion are characterized by high resolving power, so that the dependence of a particular effect on the neutron energy can be ascertained to high accuracy. The slow neutron range is conveniently subdivided into a number of sub-ranges, of which the most important are : (1) Cold Neutrons. These constitute a special category of slow neu- trons of energy less than ^0.002 ev ; they exhibit an anomalously large penetrability through crystalline or polycrystalline materials. (2) Thermal Neutrons. In diffusing through materials with relatively small neutron absorption, slow neutrons tend to assume a velocity dis- tribution of the Maxwellian form dniv) = Av 2 e- M °* /2kT dv (3) The peak of the Maxwellian distribution is at an energy E — kT, where T is the absolute temperature of the medium through which the neutrons are diffusing. Neutrons having such a velocity distribution are referred to as thermal neutrons. Frequently the neutrons diffusing through a given medium are not in thermal equilibrium with the medium. This results in a rather greater preponderance of high-energy (slow) neutrons than is given by the Maxwell distribution function. Slow neutrons originating from such a source, generally of energy above ~0.5 ev, are often referred to as epi- thermal neutrons. In much of the work on slow neutrons the thermal neutrons are separated from neutrons of higher energy (say epithermal) by taking advantage of their strong absorption in relatively thin layers of cad- mium. Those neutrons in a given distribution which are absorbed by 224 The Neutron [Pt. VII a cadmium layer are sometimes called C-neutrons; the neutrons which penetrate cadmium (energy > 0.3-0.5 ev) are, correspondingly, called epicadmium neutrons. (3) Resonance Neutrons. This classification refers to slow neutrons of energy between ~1 and ~100 ev; it is based on the large number of distinct, sharp, absorption resonances which have been observed in the interaction of neutrons, in this energy range, with heavy nuclei. (b) Intermediate Neutrons. In the intermediate range the predomi- nant type of neutron reaction is elastic scattering. Until fairly recently this region has been the least extensively studied, mainly because of the lack of suitable neutron sources and detectors. Recently a number of techniques have been developed for the study of this energy range. These techniques are now being extensively exploited. (c) Fast Neutrons. The fast-neutron region is characterized by the appearance of many nuclear reactions which are energetically impossible at lower neutron energies, of which the most important is inelastic scat- tering. This region has been fairly extensively investigated, although the available techniques have been rather crude as compared to those used in the slow-neutron range. (d) Very Fast Neutrons. This energy interval is distinguished from the preceding by the appearance of nuclear reactions involving the emis- sion of more than one product, such as the (n,2n) reaction. It is a rela- tively unexplored energy range, mainly because of the comparatively small number of suitable neutron sources. (e) Ultrafast Neutrons. The development of ultrahigh energy particle accelerators has resulted in the possibility of producing ultrafast neutrons and studying their properties. Before the advent of these machines, this region was accessible only through the utilization of the neutrons in the cosmic radiation. A distinguishing feature of nuclear reactions in this energy range comes from the relatively small interactions of neutrons with nuclei, resulting in a partial transparency of nuclei to ultrafast neutrons. Also, for these high energies, "spallation reactions" — in which the bombarded nucleus emits many fragments — are observed. C. Fundamental Interactions Of primary importance for the understanding of nuclear forces is the study of the interaction between neutrons and other nucleons — neutrons and protons — and between neutrons and the lighter particles (electrons, mesons) . There have been many experimental and theoretical investiga- tions which shed light on the nature of nuclear forces. Among the most important are the scattering of neutrons (and protons) by protons, the Sec. 1C] Properties and Fundamental Interactions 225 capture of neutrons by protons, the static and dynamic properties of the deuteron, the scattering of neutrons by hydrogen molecules, and some of the properties of heavy nuclei. Since most of these results have already been discussed in Part IV of Volume I, they will merely be presented here in summary form. However, since a number of aspects of the neutron-proton interaction are of primary importance for the understanding of the interaction of neutrons with matter, such aspects will be discussed more fully. Thus, the scattering and capture of neutrons by protons will be covered in some detail. The discussion of the scattering of neutrons by hydrogen molecules, involving as it does the wave properties of neutrons, will be reserved for Section 5. 1. The Neutron-Proton Interaction, (a) Properties of the Deuteron. (1) Static Properties. The ground state of the deuteron is essentially a 3 S% configuration. 1 From the known binding energy of the deuteron, conclusions can be drawn concerning the force (potential) between the neutron and proton in this state, in which the neutron and proton spins are aligned in the same direction (total spin = spin of the deuteron = 1). The static properties of the deuteron give no information con- cerning the neutron-proton potential in the singlet state (total spin = 0) (B22). (2) Photodisintegration. The deuteron can be disintegrated into a neutron and a proton by the absorption of an amount of energy greater than its binding energy, 2.23 Mev. Although this can be accomplished through the bombardment of deuterons by a variety of particles, the disintegration of the deuteron was first observed (C8) in bombardment by gamma radiation. This process, the so-called photodisintegration of the deuteron, has been most extensively investigated, and has yielded much useful information concerning the neutron-proton interaction (B22). Immediately after its discovery, Bethe and Peierls (B20) and Massey and Mohr (M6) propounded the theory of the disintegration of the deuteron by a photoelectric effect. Soon afterward, Fermi (F15) sug- gested a second mechanism which contributes to the disintegration: a photomagnetic process. In the photoelectric disintegration the electric field of the gamma-radiation acts on the instantaneous electric dipole moment of the deuteron (the average electric dipole moment is, in the 1 The deuteron has a small electric quadrupole moment, which fact implies that the ground-state wave function contains, in addition to the predominant 3 Si func- tion, a small admixture of 3 Z>i form. The implications of this fact, especially with regard to the tensor nature of nuclear forces, has been discussed in detail in Part IV of Volume I. 226 The Neutron [Pt. VII ground state, zero). After separation of the proton from the neutron, through the action of the electric field, the spins of the neutron and proton remain parallel; thus, the photoelectric effect involves only the properties of the neutron-proton interaction in the triplet state. In the second process the magnetic field of the radiation interacts with the magnetic dipole moments of the neutron and proton, the effect of the interaction being to "flip" the spin of one of the particles with respect to that of the second; thus, the photomagnetic process also involves the properties of the neutron-proton potential in the singlet state. The contributions of the two processes can be separated experimen- tally, since they result in different angular distributions of the emerging particles with respect to the direction of the incident gamma-ray. These angular distributions can be understood in terms of a rough, semi-classi- cal model of the deuteron as a neutron and proton separated by a fixed distance and having equal probability for all possible orientations of the connecting line. Thus, in the case of photoelectric disintegration, the effect is greatest when the deuteron's dipole moment is in the direction of the electric field vector of the gamma-ray beam, which is perpendicu- lar to the direction of motion of the photon. This leads to a distribu- tion of recoil neutrons and protons (in the center-of-mass coordinate system) proportional to the square of the sine of the angle between the incident photon and the recoils. In the photomagnetic disintegration, on the other hand, the effect is produced by the difference between the action of the photon's magnetic field on the neutron and the proton, which is due to the difference be- tween the magnetic dipole moments of the two particles [—1.91280 and +2.79255 nuclear magnetons, respectively (Ml)]. This difference is independent of the orientation of the deuteron, and hence the recoils, which are due to the photomagnetic disintegration, are spherically sym- metrically distributed in the center-of-mass coordinate system. 1 In addition to the different angular distributions, the cross sections for the photoelectric and photomagnetic disintegrations have different dependences on the gamma-ray energy. The cross sections start out at zero at the photodisintegration threshold, rise to maxima for values of 1 This difference in the angular distributions of the recoil products follows from simple quantum-mechanical considerations. Thus, in the photoelectric effect the final state of the separated neutron and proton must be a P state to satisfy the selection rule for electric dipole radiation, Ah = ±1; also, since the electric field does not act, in first order, on the nucleon spins, AS - 0. The transitions occur from the state of the deuteron for which m = ±1 (with respect to the direction of the incident photon); hence the sin 2 distribution of the recoils. Correspondingly, the magnetic dipole transition requires AL = 0, AS = ±1. Thus, the final state is a 1 S a state, with a spherically symmetrical distribution of the recoils. Sec. 1C] Properties and Fundamental Interactions 227 E y = — 2.2 Mev of approximately twice the binding energies (of the trip- let and singlet states, respectively), and then fall off with increasing gamma-ray energy. 1- For photon energies close to the threshold, the two effects are comparable in importance. For photon energies large compared to the binding energy of the deuteron, the photomagnetic effect is negligible compared to the photoelectric effect. Thus, the angular distribution of the recoils can be described by the expression a + b sin 2 0, with a and b both energy-dependent, and a falling off much more rapidly than b with increasing gamma-ray energy. For photon energies considerably above the threshold, the simple considerations described above are no longer completely applicable; other effects, such as the tensor nature of the neutron-proton interaction, come into play (R6). Since the early attempts to observe the cross section and angular distribution of the recoils in the photodisintegration of the deuteron (CIO), the techniques of measurement have been considerably improved. More recent results (B30, G28, W44) confirm the theoretical predictions and provide important information concerning the nature of the neutron- proton interaction. (6) The Capture of Neutrons by Protons. Although, for neutrons of most energies, the most important effect in the neutron-proton interac- tion is the scattering process, discussed in (c) below, it was soon observed (W18, D19, A20) that neutrons are appreciably absorbed in hydrogen- containing substances. This absorption is attributable to a radiative capture (n,y) process, a reaction to be expected, since the combination of a neutron and a proton into a deuteron is energetically preferred. The gamma-rays accompanying this process were first observed by Lea (L10). The radiative capture reaction, n + p — > d + 7, is the inverse of the photodisintegration of the deuteron, and the calculation of its cross section follows directly from the photodisintegration calculations (B22). Such a calculation indicates that capture of a slow neutron by a photo- electric process (emission of an electric dipole gamma-ray) has a negligi- ble cross section (B20) ; this result is easily understood, since photoelec- 1 If (3 = 2.2 Mev and | n | = 0.065 Mev are the absolute values of the binding energies in the triplet and singlet states (the singlet state is actually unbound) and E = E y — €3, then the simple theory gives (B22) &re 2 h 2 t3 A E* ff mag — 3 he M (E + e 3 ) s 2ir e 2 h 2 Qfr - ;Q 2 («* + 1 q |*)»«,*g* 3 he M Mc 2 (E + caKE + |ta I) 228 The Neutron [Pt. VII trie capture requires that the neutron-proton system be, initially, in a P state, which is highly improbable for a slow neutron. Indeed, it was the observation of an appreciable neutron-proton capture cross section which led Fermi (F15) to postulate a photomagnetic capture process and its inverse, the photomagnetic disintegration. Since the photo- magnetic capture involves only S states C-S —> s Si), the resulting cross section has al/» dependence on the neutron energy and therefore becomes quite appreciable for thermal neutrons. Fermi showed that the photomagnetic capture is indeed strong enough to account for the relatively rapid absorption of thermal neutrons in hydrogenous materials. The precise experimental determination of the neutron-proton (n,y) cross section is difficult, since capture represents only a small fraction of the total neutron-proton cross section at even the smallest available neutron energies and, further, it leads to a non-radioactive end product. Thus, none of the usual methods of measuring cross sections (Section 3) is easily applicable. Its value can, however, be inferred from measure- ments involving the diffusion of thermal neutrons in hydrogen-contain- ing materials (Section 4), since the average distance traveled, before absorption, by a thermal neutron depends on the capture as well as on the scattering cross section (A20). Or, equivalently, the mean life of a thermal neutron in a hydrogenous material is inversely proportional to the capture cross section, so that a measurement of the mean life can yield a value for <r(n,y) (A18, M4). The most accurate value of the (n,y) cross section has been obtained from a direct comparison of slow neutron-proton absorption with the absorption of boron, which also has a 1/v energy dependence (R8, W19). Whitehouse and Graham (W19) obtained — = 2270 ± 68 Using the value <tb = 710 ± 21 barns for neutrons of velocity v = 2200 m/sec (R20), this yields <m(n,y) = 0.313 ± 0.013 barns Since the photomagnetic capture and disintegration processes involve the properties of the neutron-proton interaction in the singlet state, the magnitudes of the cross sections can be used to infer the strength of this interaction. In particular, different cross-section values are predicted, depending on whether the singlet state is bound or unbound (B22). In the case of the photodisintegration experiments, the cross-section deter- minations are not sufficiently accurate to allow an unambiguous choice between the two possibilities (G28). Sec. 1C] Properties and Fundamental Interactions 229 The neutron-proton radiative capture cross section is measured with sufficient accuracy. Rosenfeld (R19) calculates the cross section, for neutrons of energy 0.026 ev, to be 0.32 or 0.16 barn for the cases, respec- tively, of an unbound or a bound singlet state. The experiments strongly favor the conclusion that the ^o state of the deuteron is un- bound. "But [chiefly because of the neglect of the exchange effect] the calculation is not sufficiently accurate to allow us to regard this evidence as entirely conclusive." The conclusion is, however, completely borne out by the measure- ments on the scattering cross sections for ortho- and parahydrogen (see Section 5). (c) The Scattering of Neutrons by Protons. (1) Cross Section. Since the first investigations of Chadwick (C7), there have been many meas- urements, at various neutron energies, of the cross section for the scat- tering of neutrons by protons. The results are summarized in Fig. 2, _, 2 M a u t S b 6 4 H 0.03 23468 23468 28468 23468 2 0.01 0.1 1 10 100 300 E n (Mev) Fig. 2. Cross section for the scattering of neutrons by free protons vs. kinetic energy of the neutron, in the range 0.01-300 Mev, from the compilation of Adair (A2). Additional data, in the ultrafast range, are given in (T3) and (D12). in which the neutron-proton scattering cross section is plotted as a func- tion of the neutron energy, between 0.01 and 300 Mev. The quantum-mechanical treatment of neutron-proton scattering was first given by Wigner (W27), who showed that the cross section as a 230 The Neutron [Pt. VII function of neutron kinetic energy E is given (in the limit of neutron wavelength large compared to the range of the neutron-proton force) by the expression „ 4irh 2 er(n-p) = ■ = : — j- (4) M(^+|c|) where e is the binding energy of the neutron-proton system (deuteron). The above expression must be corrected for the finite range of the neu- tron-proton force (B22); however, this correction is independent of E and need not be considered for the purposes of the arguments which follow. The above expression was found to agree with the observed cross sections for neutrons of ~1 Mev and greater. When applied to slow neutrons, the above expression yields a value of ~3 barns. The early experiments (D19) gave a value, very much greater, of ~35 barns. (The cross section for the scattering of neutrons by free protons has, in fact, been determined to be ~20 barns.) This serious discrepancy was explained by Wigner x under the assump- tion that the force between a neutron and proton depends on the rela- tive orientation of their spins. Since in a fourth of the cases the neutron- proton scattering takes place in the singlet state, the cross section should be given by the expression 47rh 2 /3 1 1 1 \ c(n-p) = 1 - i r -\ — ^ i r ) (5) M \4|^+|6 3 | 4|tf+U|/ where e 3 and « x refer, respectively, to the binding energies of the triplet and singlet states of the deuteron. To fit a slow-neutron cross section of 20 barns, | ei | = 0.065 Mev. From the neutron-proton scattering, it is impossible to determine the sign of €i. Actually, we know from other evidence that the singlet state of the deuteron is unbound. Thus, although the term "binding energy" has no meaning for this state, «i represents a certain combination of the constants which describe the neutron-proton interaction in the singlet state (B22). Early measurements of the slow neutron-proton scattering cross section (D19, W18, A20) showed a wide variation, outside the experi- mental uncertainties. The reason for these variations was given by Fermi (F17), who ascribed them to the effect of the binding of the pro- 1 Although it is universally acknowledged throughout the literature (F5, B22) that Wigner originated the idea of the spin dependence of the neutron-proton force to explain the large neutron-proton scattering cross section for slow neutrons, he has not, to our knowledge, published these considerations. Sec. 1C] Properties and Fundamental Interactions 231 tons in the material — molecule, liquid, or solid — used as a neutron scat- tered Such effects are, of course, negligible for neutron energies con- siderably greater than the binding energy of the proton in the system under consideration. They are most important for neutron energies less than the lowest excitation energy of the system in which the proton is bound. The proton behaves like a particle of infinite mass when bound in solids or liquids, or like a particle of the total mass of the molecule involved in the case of neutron scattering by gaseous materials. It can be shown that the scattering cross section of slow neutrons by bound protons varies directly as the square of the reduced mass of the neutron, the scattering being spherically symmetrical in the center-of-mass co- ordinate system. Thus, the scattering cross section of infinitely slow (zero energy) neutrons by protons in solids or liquids should approach four times the free proton cross section, or about 80 barns. The variation of the neutron-proton cross section in the energy range in which the proton is neither free nor completely bound (energy greater than the lowest excitation energy but less than the binding energy of the protons) depends on the specific properties of the proton-containing material under consideration and is rather difficult to compute. Dis- regarding interference effects due to the crystalline properties of the solid under consideration or to the order introduced by the molecular structure (see Section 5), the cross section can be shown to decrease monotonically from the bound proton to the free proton cross section as the neutron energy increases from to several electron volts (A31, M27). Thus, the free proton scattering cross section for slow neutrons can be obtained from measurements on neutrons in the resonance energy range. As a result of such measurements (C18), the slow neutron-free proton scattering cross section was determined to be slightly less than 21 barns. However, binding effects still have a small influence for neu- trons with energy of a few electron volts. In the most recent experi- ments (M24), the neutron-proton cross section has been measured as a function of the neutron energy, and theoretical considerations (P19) were used to extrapolate to a free proton neutron-proton scattering cross section of 20.36 ± 0.10 barns for neutrons of zero kinetic energy. Effects of nuclear binding are, of course, present in the scattering of neutrons by heavier nuclei, although their importance decreases rapidly with increasing mass number A of the scattering nucleus. In general, for nuclei bound in solids or liquids, (A + 1\ 2 Cbound — I I ""free (6) 232 The Neutron [Pt. VII Thus, for deuterons bound in solid material, <rb OU nd = 2.25<rf re e, while, for the scattering of slow neutrons by carbon, abound = 1.17o-f ree . (2) Angular Distribution. It was first pointed out by Wigner (W27) and by Wick (W22) that the angular distribution, in the scattering of 20 - -15 - slO 1 1 1 1 1 1 ' ' 1 o """»>. „ *■"' V N °s' v s 40 Mev s s \ V \ / \ / \ / \ fa ° \ ' T \ • v / — s s ° o o" -"-'o' s'o ° =~^ n 90 Mev s -A ' N T - — tk~ \ / \- .7- V y j. 1 1 1 \260 *Xi \Mev £4rj- T - \ S^.x h ■■ 1 1 1 1 5 .y i 20 40 60 80 100 120 140 160 180 Scattering angle Fig. 3. Angular distributions in the scattering of ultrafast neutrons by protons. The figure is from a paper by Kelly, Leith, Segre, and Wiegand (K9). fast neutrons by protons, should provide information concerning the neutron-proton interaction in states of higher angular momentum (P, D, etc). The scattering of slow neutrons by protons involves only the state of zero angular momentum (S state) and is therefore spherically symmetrical in the center-of-mass coordinate system. The energies at which higher angular momentum scatterings set in measure the strengths Sec. 1C] Properties and Fundamental Interactions 233 of the interaction in these states, while the form of the angular distribu- tion leads to inferences concerning the nature of the potential. A number of experiments using d + d neutrons (2.4 to 2.7 Mev) show that, at these energies, the neutron-proton scattering is spherically sym- metrical in the center-of-mass system (R19). While the scattering is still essentially spherically symmetrical at 14 Mev (BIO), some of the experiments in the 9-14-Mev neutron energy range (L7, P24) show indi- cations of a slight asymmetry, corresponding to a preferential scattering of protons in the forward direstion. This asymmetry is quite pro- nounced for neutrons of energy 27 Mev (B71). Such an effect is to be expected on the basis of a theory of a neutron-proton interaction of the exchange type; exchange forces are required to explain the "saturation" of nuclear forces, since they lead to a repulsive force at short distances and thus tend to prevent the collapse of heavy nuclei and to maintain a constant density of nuclear matter. Neutron-proton scattering experiments with 40-, 90-, and 260-Mev neutrons show large deviations from spherical symmetry; their results are plotted in Fig. 3. The most interesting characteristic of these re- sults is the rough symmetry about 90° (in the cm. system) of the dif- ferential neutron-proton scattering cross section. Serber has pointed out that these results indicate a neutron-proton interaction involving approximately equal proportions of forces of the ordinary and exchange type. Christian and Hart (CI 5) have discussed the type of neutron-proton interaction which is required to fit both the low-energy and the high- energy data. They use an interaction potential of the Serber type, choosing the radial dependence to obtain the best possible fit, and also taking into account the tensor form of the interaction, required by the electric quadrupole moment of the deuteron. The Serber interaction, a mixture of equal parts ordinary and exchange forces, has the property that all terms which correspond to odd orbital angular momentum quantum numbers vanish; the observed symmetry of the high-energy neutron-proton scattering about 90° is, of course, the decisive evidence in favor of the Serber interaction, since any appre- ciable interaction in a state of odd orbital angular momentum would destroy this symmetry. However, the terms of odd angular momentum are precisely those which lead to a strong repulsion between the neutron and proton for small distance of separation and, therefore, are required for the "saturation" of nuclear forces. Indeed, it appears doubtful whether a nucleon-nucleon interaction of the Serber type is consistent with the saturation of nuclear forces in heavy nuclei. 234 The Neutron [Pt. VII 2. The Neutron-Neutron Interaction, (a) Experimental Evidence. (1) Low Energies. Since the highest available neutron densities are still far from appreciable (below 10 10 neutrons/cm 3 ), it is manifestly im- possible to obtain information about the neutron-neutron interaction through observation of the scattering of neutrons by free neutrons; our knowledge of the force between two neutrons must, perforce, be inferred from other information. In considering the available information, it is useful to bear in mind the following general considerations: (1) Experi- ments involving neutrons of kinetic energy below ~20 Mev yield infor- mation only about the neutron-neutron forces in the singlet S state since, at these energies, states of higher angular momentum are not appreciably excited, and the exclusion principle prevents the neutron-neutron system from existing in a triplet S state. Experiments with ultrafast neutrons can, on the other hand, yield information about the neutron-neutron interaction in both the singlet and triplet states. The same arguments hold, of course, for the proton-proton interaction. (2) Since there is some reason to expect that the purely nuclear interaction between two nucleons should be independent of whether they are neutrons or pro- tons (see further on), the evidence on the proton-proton interaction can be considered to have some bearing on the problem of the neutron- neutron interaction. In any event, we shall herein review, briefly, the available information on the interaction between two nucleons, irrespec- tive of their charge. 1 The strongest evidence concerning the charge symmetry of nuclear forces (equality of the nuclear neutron-neutron and proton-proton forces) is derived from the observed energy differences between mirror- image nuclei (nuclei which can be obtained from one another by inter- changing the neutrons and protons). The energy differences between such nuclei, as measured by the maximum energy of the beta-rays emitted in the decay of one of the pair into the other, can be completely accounted for by the neutron-proton mass difference and the difference in the electrostatic energies of the two nuclei. In addition, mirror nuclei are similar with respect to their observed level structure. From this evidence, we conclude that the purely nuclear neutron-neutron and proton-proton forces are essentially equal. Actually the evidence on nuclear level structure lends strong support to the stronger hypothesis of charge independence (equality of the neutron-neutron, neutron- proton, and proton-proton forces in the same states), since nuclear iso- bars are observed to exhibit markedly similar level patterns. 1 A much more complete discussion of these points is contained in Part IV of Vol- ume I, Sec. 1C] Properties and Fundamental Interactions 235 The purely practical restrictions, which prevent the performance of neutron-neutron scattering experiments, do not obtain in the case of proton-proton scattering. Many such experiments have been performed for proton energies up to <~15 Mev. From these it can be concluded that within the experimental uncertainty (and taking into account the effect of the Coulomb interaction) the nuclear neutron-proton and pro- ton-proton forces in the 1 S state appear to be equal. Thus, the available evidence supports the idea of the charge inde- pendence of the nuclear forces between two nucleons. (2) On the Possible Existence of a Stable Di-Neutron. The strength of the neutron-neutron interaction in the '(So state determines whether or not there can be a stable system consisting of two neutrons. (There is no possibility for a stable two-nucleon state of higher orbital angular momentum.) The existence of such a stable "di-neutron" is, until now, experimentally neither proved nor disproved. If the nucleon-nucleon interaction is completely charge independent, the di-neutron will be unstable, since the '(So neutron-proton system is unstable by about 65 kev; however, this is so close to being stable that it would require only a rather small increase of the neutron-neutron interaction over the corresponding neutron-proton interaction to lead to a stable di-neutron. A stronger argument against the existence of a stable di-neutron can be derived from the equality of the low-energy neutron-neutron and proton-proton forces, as evidenced by the properties of the mirror-image nuclei. Analysis of low-energy proton-proton scattering (Jl) proves, conclusively, that the '£ proton-proton interaction is not strong enough to lead to a bound state (even in the absence of the Coulomb repulsion). Although the weight of available evidence does not favor the existence of a stable di-neutron, it is, nevertheless, of interest to consider some of the possible consequences of its existence. Feather (F4) has pointed out that it is possible to obtain an upper limit to the binding energy of the di-neutron from the observation that the nucleus He 6 does not decay according to the reaction He 6 -»• He 4 + n 2 . From the masses of the nuclei involved, he concludes that the binding energy of the di-neutron is less than 0.7 ± 0.2 Mev. He also points out that the di-neutron would be beta-unstable, n 2 -» H 2 + P~ + v, with a mean life (assum- ing an allowed transition) of 1 < t < 5 sec. If the di-neutron should exist as a stable configuration, and if it should be possible to obtain it in sufficient numbers, it could easily be detected in experiments involving scattering by nuclei. For the two neutrons would be scattered coherently, and the interference effects would per- sist to considerably higher energies than do the normal coherence effects in slow-neutron scattering, since the di-neutron would be a compact sys- 236 The Neutron [Pt. VII tem, of dimensions ~10 -12 cm. Thus, the scattering of di-neutrons by hydrogen atoms would be the same as the scattering of cold neutrons by parahydrogen molecules (see Section 5), for which the cross section is ~4 barns, as compared to ~20 barns for the free neutron-proton scat- tering cross section. Furthermore, as pointed out by Feather, capture of a di-neutron would, as compared to ordinary neutron capture, result in a different compound nucleus and in characteristic radioactivities. The difficulty in observing such effects would, if the di-neutron were stable, arise from the difficulty of obtaining an appreciable source of di-neutrons. Such a source would have to be obtained from a suitable «.uclear reaction, in which di-neutrons are emitted. It could not be ob- tained by neutron-neutron collisions in a region of high slow-neutron density (even if neutron densities of sufficient magnitude were avail- able) because the required reaction, n + n — > n 2 + 7, would be highly forbidden, since it would involve the transition 1 S — * 1 £>o- Although a di-neutron could, if stable, be produced in a collision involving three neutrons, such collisions would be exceedingly improbable. Fenning and Holt (Fll) have attempted to detect the presence of di-neutrons in the Harwell pile, by looking for alpha-particles from the Bi 209 + n 2 -> AcC 211 — ^ AcC" -?-> Pb 207 2.16 min 4.8 min (They were investigating the possibility that di-neutrons might be emit- ted in slow-neutron fission.) They exposed bismuth to a flux of ~10 12 neutrons -cm -2 -sec -1 , but could detect no activity attributable to AcC, from which they established an upper limit of 1.5 X 10 -21 sec -1 for the product of the di-neutron flux and the cross section for its absorp- tion in bismuth. In considering the possible effects of the di-neutron on nuclear reac- tions, it should be noted that such effects might exist even if the di- neutron is not stable, for the dynamics of a reaction involving two neu- trons would be quite different if the two neutrons were absorbed or emitted as a single unit than if they behaved independently (C19). Thus, Kundu and Pool (K25) consider the characteristics of the excita- tion of (H 3 ,p) reactions on rhodium and cobalt as "probable evidence of the di-neutron." Another, and more favorable, reaction in which effects of a di-neutron might be observed involves the two possibilities H 3 + H 3 -» He 4 + 2n + Q t -> He 4 + n 2 + Q 2 If the two neutrons are emitted independently, their energies, and that Sec. 1C] Properties and Fundamental Interactions 237 of the recoil alpha-particle, will vary over a rather wide range (for a given energy of the bombarding triton), as is to be expected in a three- particle reaction. If, on the other hand, the two neutrons are emitted as a single particle, the energies of the di-neutron and of the alpha- particle are uniquely determined, for a given angle of emission, by the laws of conservation of energy and momentum. The weight of available evidence favors the first of the two above-mentioned reactions (L22, A9, L14). Perhaps the most striking evidence concerning the di-neutron is derived as a by-product of the experiments of Panofsky, Aamodt, and Hadley (P3) on the absorption of negative pi-mesons by deuterium in the reaction ir~ + d — » In + y. In interpreting the observed gamma- ray energy distribution, Watson and Stuart (W6) point out that the shape is strongly dependent on the degree of correlation in the direc- tions of emission of the two neutrons; indeed, if the neutrons were always emitted as a di-neutron, the gamma-ray spectrum would be monochro- matic. The observations are sufficiently accurate to show that there is a strong interaction (attraction) between the two neutrons; they are not sufficiently precise, as yet, to allow more than an upper limit of <~200 kev to be placed on the binding energy of the di-neutron. The proper- ties of the meson-producing reaction, 7 + d — » 2n + x + , might also throw some light on the di-neutron. (3) High Energies. For neutrons with kinetic energies in the ultra- fast range, it becomes possible to observe the neutron-neutron interac- tion somewhat more directly in the scattering of neutrons by deuterium nuclei. If the wavelength of the neutron is much less than the average separation between the neutron and proton in the deuteron, the two nucleons should scatter neutrons independently; the cross section for the scattering of neutrons by neutrons, a(n-n), should then be given by the difference between the scattering cross section of deuterium and that of hydrogen, ,_ e{n-n) = <j(n-d) — <r(n-p) V) Unfortunately, there are still appreciable effects of the structure of the deuteron in the scattering of neutrons of a few hundred Mev, since the neutron wavelength is not negligible compared to the internucleon distance. Another way of expressing this difficulty is to note that the nucleons in the deuteron are moving with kinetic energies of internal motion as high as <~25 Mev; the velocity, corresponding to this kinetic energy, is only ~i of the velocity of a 100-Mev neutron. A possible way out of this difficulty has been suggested by Segre. 1 Since it is feasible to measure, directly, both a(n-p) and <r(p-p), a meas- 1 Private communication. 238 The Neutron [Pt. VII urement of o-(p-rf) should determine the magnitude of the effect, on the cross section, of the structure of the deuteron. This correction could then be applied to the measurement of o(n-d) in order to extract the neutron-neutron cross section. The interpretation of neutron-deuteron scattering for ultrafast neu- trons has been considered by a number of authors, e.g., De Hoffman (D8), Gluckstern and Bethe (G9), and Chew (C14). It turns out that, in addition to the expected contributions from neutron-proton and neutron-neutron scattering (both averaged over the momentum dis- tributions of the nucleons in the deuteron), there is a cross term due to interference between the two scattering centers and depending in de- tail on the nature of the interactions. As a result, the simple additivity relationship of Eq. (7) cannot even be applied at energies of a few hun- dred Mev. The most extensive attempt at interpretation of neutron-deuteron and proton-deuteron scattering experiments in the ultrafast energy range has been carried out by Chew (C14). He has treated the problem by the "impulse approximation," in which the scattering nucleon is as- sumed to behave in a relatively independent fashion during the time of the impact. This approximation appears to be appropriate for the treatment of some aspects of the neutron-deuteron scattering problem, e.g., the dissociation, by neutron impact, of the deuteron; however, other aspects, such as elastic neutron scattering or proton "pick-up," cannot be treated in this fashion. From the available data, Chew con- cludes that there is "no evidence for a difference between neutron- neutron and proton-proton interactions." The results of measurements, on the cross sections for scattering of ultrafast neutrons by protons and deuterons, are summarized in Table 2. 1 TABLE 2 Total Cross Sections of Protons and Deuterons for Ultrafast Neutrons cr(n-d) <rin-p) Difference E n (Mev) (barns) (barns) (barns) Reference 42 0.289 ±0.013 0.203 ± 0.007 0.086 ±0.015 H61 85 0.117 ±0.005 0.083 ±0.004 0.034 ±0.006 C25 95 0.104 ±0.004 0.073 ±0.002 0.031 ± 0.004 D9 270 0.057 ±0.003 0.038 ± 0.002 0.019 ±0.003 D10 280 0.049 ±0.005 0.033 ±0.003 0.016 ± 0.006 F41 (b) Some Comments Pertaining to the Meson Theory of Nuclear Forces. On the basis of the observed properties of the forces between two nu- cleons (in particular, the short range), Yukawa predicted the existence 1 For results of p-p scattering measurements see Volume I, Part IV, Section 2A. Sec. 1C] Properties and Fundamental Interactions 239 of a particle of mass intermediate between that of the electron and the proton. According to his idea the force between two nucleons is regarded as resulting from the interchange of such particles, called mesons, be- tween nucleons, in much the same way that purely electrical forces may be regarded as resulting from the interchange of photons between charged particles. There have been developed, since Yukawa's sugges- tion, a number of different types of meson theories of nuclear forces; these differ in the properties of the assumed mesons and in the type and strength of the assumed coupling between the meson and the nucleon fields (P7). Yukawa's prediction was soon followed by the discovery of such a particle in the cosmic radiation. This particle, the mu-meson or muon, has a mass of ~210 electron masses, may be either positive or negative, and is unstable, decaying into an electron and two neutral particles of negligible rest mass (presumably neutrinos) with a mean life of 2.15 X 10 — 6 sec. However, subsequent investigation has shown that the muon has a very weak interaction with nucleons, a fact which eliminates it from the role of nuclear binding material. There is, however, another type of meson, the pi-meson or pion, which appears to be more directly associated with the nuclear forces. It has a mass of ~275 electron masses, and decays into a muon and (presum- ably) a neutrino, with a mean life of 2.6 X 10~ 8 sec. Pions have been produced in the interaction between two nucleons, or between a nucleon and a gamma-ray, when there is enough energy available to supply the rest mass. In addition to charged pions of both sign, there have also been observed neutral pions (whose mass appears to be slightly less than the mass of the charged pion) which decay into two gamma-rays, with a very short mean life (~10 — 15 sec). Finally, there is evidence for the existence of one or more types of still heavier meson. Very little is known concerning their interactions with nuclei. Of the various types of mesons which have been observed, it is the pion which most probably plays the role of nuclear binding agent. It is produced directly in the interaction between two nucleons, and between nucleons and gamma-rays; the cross section for its production is of an order of magnitude which indicates that it interacts strongly with nu- cleons — strongly enough to account for the strength of nuclear forces. Indeed, recent work on the production of pions, through the use of ultrahigh-energy machines, and on the interaction of pions with nucleons and nuclei provides the strongest arguments for the fundamental valid- ity of the meson theory of nuclear forces. Although meson theories have, so far, been unable to account in detail for the nuclear forces, they do provide a qualitative basis for under- 240 The Neutron [Pt. VII standing the properties of nucleons (like, say, the anomalous magnetic moments of the neutron and proton) and of nuclear forces. The evi- dence on meson production at high energies together with the evidence on the charge independence of the nucleon-nucleon forces favor a symmetrical meson theory, in which charged mesons of both sign, as well as neutral mesons, play a comparable role. The exchange of charged mesons leads to the neutron-proton force, while neutral mesons are ex- changed in the neutron-neutron and proton-proton interactions. 3. The Neutron-Electron Interaction. From the observation that the neutron has an intrinsic magnetic moment, it may be inferred that the neutron is a complex particle whose structure contains, at least for part of its existence, some moving charges. Indeed, according to current meson theories, the neutron can dissociate, spontaneously, into a tightly bound system of proton and negative pion, according to the reaction n +±p + 7T — During the fraction of its existence as a neutron proper, it can be re- garded as truly neutral; in the proton-pion state, however, the neutron has a charge structure corresponding to a relatively concentrated posi- tive charge surrounded, at a somewhat greater distance (of the order of a few times 10 — 13 cm), by a cloud of equal negative charge. The frac- tion of the time during which the neutron is in this dissociated state is variously estimated in the different meson theories to lie somewhere in the range ~0.1 to 0.5. As a consequence of this structure, there is an electromagnetic inter- action between neutrons and electrons, the predominant aspects of which can be described in terms of the interaction between two magnetic dipoles (that of the electron and that of the neutron). This magnetic dipole-dipole interaction is, of course, strongly dependent on the rela- tive orientations of the spins of the particles. It has been investigated extensively in studies of the scattering of slow neutrons by paramagnetic and ferromagnetic materials; these phenomena will be discussed in de- tail in Section 5. A magnetic interaction of the dipole-dipole type would, because of its spin-dependent character, vanish for the case of a neutron interacting simultaneously with an even number of electrons which are in a 1 »So state. Nevertheless, it has been observed that there is a small, spin- independent neutron-electron interaction. Such an interaction follows from meson-theoretic descriptions of the structure of the neutron, and its observed magnitude is in rough agreement with theoretical expecta- tions (to be discussed later in this section). The strength of the inter- Sec. 1C] Properties and Fundamental Interactions 241 action is such that it leads to a (spin-independent) cross section for the scattering of neutrons by a single bound electron of ir,~5X 10~ 7 barns. Although the value of <r e is exceedingly small as compared to the cross sections for scattering of neutrons by nuclei (~ a few barns), effects of this scattering are, nevertheless, detectable in the scattering of slow neu- trons by some heavy atoms. The possibility of observing such effects was first suggested and discussed by Condon (C24). This possibility results from the interference between the elastic scattering of sufficiently slow neutrons by the nucleus and by all the electrons in the atom. Let us consider the scattering of a beam of slow neutrons, wavelength of the same order as the atomic dimensions, by a single atom containing Z electrons. The differential scattering cross section is -£ = | a + Za e F(6) \ 2 (8) where «o an d a e are, respectively, the amplitudes for scattering of a neu- tron by the nucleus and by a single bound electron, 1 i.e., <r e = 4ira e 2 . F(d) is the atomic electron form factor, which can be computed from the electron distribution or, alternatively, obtained from data on the scat- tering of x-rays (C22). Since c — 4ira 2 5s> <r e , we may write For neutrons of wavelength large compared to the atomic dimensions, F(0) — > 1, while, as the neutron wavelength becomes small compared to the distance between the electrons, F(6) — > 0. To observe a possible neutron-electron interaction of the magnitude expected, it is absolutely necessary to take advantage of the interference between the neutron scattering by the nucleus and that by the atomic electrons. For an estimate of the magnitude of possible effects, let us assume a set of arbitrary, but reasonable, values, a ~ 5 barns, cr e ~ 5 X 10~ 7 barns, ZF « 50. Then — = 2ZF ( — ) ~ 3 percent 00 wo/ 1 Strictly speaking, ao refers only to the coherent portion of the nuclear scattering amplitude. When there is, in addition, an incoherent contribution to the nuclear scattering, the above expression for the differential atomic scattering cross section requires the addition of an incoherent term, ai 2 , where the total nuclear scattering cross section is <r s = 4ir(ao 2 + ai 2 )- Although, for the discussion of this section, we shall assume that oi = 0, all the expressions to be derived can easily be modified if the nuclei exhibit any incoherent scattering. 242 The Neutron [Pt. VII Although it should be possible to measure an effect of this magnitude, it is necessary to devise some means of distinguishing it from the purely nuclear scattering, whose cross section is seldom known to an accuracy of 3 percent. One possibility is the method devised by Fermi and Mar- shall (F23). They compared the scattering of thermal neutrons by gaseous xenon atoms, into a given solid angle, at angles of 45° and 135°. Such a difference would arise from two sources: (1) the difference in the electron form factor which was, for the neutrons employed, F(45°) — F(135°) = 0.261; (2) an asymmetry in the scattering due to the center-of-mass motion of the scattering system, since the angles were fixed in the laboratory system. In order to compute the magnitude of this effect, it is necessary to take into account the thermal motion of the xenon atoms (Doppler effect), as well as the energy distribution in the slow-neutron beam and the variation of the sensitivity of the de- tector with neutron energy. Fermi and Marshall observed a definite asymmetry in the scattering of ~2 percent which, however, became much less than the experimental uncertainty of 0.85 percent after they applied the Doppler effect cor- rection. Thus, although the accuracy of their measurement is insuffi- cient to confirm the existence of the neutron-electron scattering, their results set an upper limit to the neutron-electron interaction. If the interaction is described in terms of a fictitious square well potential of depth V and range equal to the classical electron radius, r = e 2 /mc 2 = 2.8 X 10~ 13 cm, the experiment of Fermi and Marshall gives | V Q \ <, 5000 ev. (Compare with V <~ 25 X 10 6 ev for the neutron-proton potential.) l The method of Fermi and Marshall has been reapplied by Hamermesh, Ringo, and Wattenberg 2 (H28) with, however, significant improve- 1 The spin-independent neutron-electron interaction is, if it is of mesonic origin, a short-range interaction. Consequently, the value of <r e for slow neutrons depends only on the volume integral of the interaction potential, rather than on any details of its shape. The Born approximation yields AW fl-fr where M is the neutron mass and f4w 6 = 4x Cv(r)r 2 dr = - (~\ TW for the fictitious square well potential of range ro. A value of Vo = 5000 ev corre- sponds to a e = 4.3 X 10" 7 barns. It should be noted that a negative b (attractive potential) corresponds to a negative a,., while the nuclear scattering amplitude, Oo, is usually positive. 2 I am indebted to Dr. A. Wattenberg for advance communication and discussion of these results. Sec. 1C] Properties and Fundamental Interactions 243 ments in the geometry and a large increase in the number of neutrons counted. They have obtained the value V = 4100 ± 1000 ev This value is based on measurements of the scattering of krypton as well as of xenon; measurements were also performed on argon to check the correction for the center-of-gravity motion. Of the uncertainty of ±1000 ev, quoted by these investigators, half is statistical and half is their estimate of possible systematic errors. Their result is in good agreement with the previous measurement of Havens, Rabi, and Rain- water (H46, H47). The method of observing the neutron-electron interaction, devised by Havens, Rabi, and Rainwater, is based on observation of the total cross section = fda (6) = <r ± 2ZF(cr cr e y A (9) They distinguish the effect of the coherent electron scattering by ob- serving the variation of a with neutron energy for wavelengths of the order of the atomic dimensions. For a neutron source of variable energy, they employed the Columbia velocity selector; as scattering materials, they used liquid lead and bismuth. Their method suffers from the difficulty that it is necessary to apply rather large corrections to the observed cross section vs. energy curve before the effect of the neutron-electron interaction can be ascertained. These corrections arise from a number of causes: (1) In addition to the nuclear scattering there is, for lead and bismuth, a small nuclear absorption whose cross section varies with the neutron velocity as 1/v. This correction can be applied with good accuracy. (2) Since they were dealing with atoms bound in a liquid, rather than with free atoms, it is necessary to take into account the effects of the binding (see page 231) on the variation of the cross section with energy. Although such effects (which include, of course, a Doppler effect for the motion of the atoms in the liquid) de- pend, to a certain extent, on the details of the binding mechanism, it has been shown by Placzek (P19) that they can be evaluated with the requi- site accuracy. (3) There are, in the scattering of neutrons by liquids, effects due to interference between the scattering by different atoms. Although such effects are relatively small, since the experiments involve neutrons with wavelengths of the order of atomic dimensions and hence somewhat smaller than the interatomic distances, they cannot be neg- lected in comparison with the effects of the neutron-electron scattering. The liquid coherence effects can be shown to vary as X 2 , in the energy 244 The Neutron [Pt. VII region of interest, and the coefficient of the X 2 term can be evaluated by a general method for systems of high density and small compressibility, as shown by the calculations of Placzek, Nijboer, and van Hove (P21). (4) Although the three effects discussed above, all of which arise from nuclear causes, can be taken into account in a satisfactory fashion, there is a possible electronic effect which must be taken into consideration. Even though the ground states of atomic lead and bismith are 1 S , and Fermi and Marshall (F23) have proved that scattering due to the neu- tron magnetic moment is negligible for such atoms, small magnetic scattering effects might possibly be present when these atoms are in the liquid state. The results upon which Havens, Rabi, and Rainwater base their value of the neutron-electron interaction were obtained by careful analysis of data on the scattering of monoenergetic slow neutrons by liquid bismuth; they lead to V = 5300 ± 1000 ev Of the quoted uncertainty, 650 ev is statistical in origin, the rest being an estimate of possible systematic errors. 1 Although a detailed meson-theoretic discussion of the spin-independ- ent neutron-electron interaction is outside the province of this review (even assuming that an adequate theory were available), the phenomena which can give rise to such an interaction are, however, susceptible of relatively simple physical interpretation. 2 Actually, there are two dis- tinct types of interaction which, together, can probably account for the observed effect. The first may be regarded as an electrostatic interac- 1 D. J. Hughes has reported (at the February 1952 meeting of the American Phys- ical Society) a measurement of the neutron-electron interaction by observation of the critical angle for total reflection of cold neutrons from a liquid oxygen-bismuth interface (see Section 5). His preliminary results are in excellent argeement with the measurements of Havens, Rabi, and Rainwater and of Hamermesh, Ringo, and Wattenberg. However, this method appears to be capable of yielding the most accurate value of the strength of the neutron-electron interaction. Using this method, Harvey, Hughes, and Goldberg have obtained Vo = 4250 ± 400 ev (pri- vate communication, October 1952). 2 Purcell and Ramsey (P32), in their discussion of the possible existence of a neu- tron electric dipole moment, point out that the observed neutron-electron interaction could, alternatively, be accounted for by the assumption of a small electric dipole moment of the neutron. The magnitude required to account for the observed inter- action is that of two opposite electron charges separated by a distance of 3 X 10 -18 cm. Since the experiments of Purcell, Ramsey, and Smith (S40) have yielded an upper limit of two electron charges separated by 5 X 10 -21 cm for the electric dipole moment of the neutron, the explanation of the observed neutron-electron interaction lies, in all likelihood, along the meson-theoretic lines indicated below. Sec. 1C] Properties and Fundamental Interactions 245 tion which arises from the fact that the neutron is not a strictly neutral particle but behaves, at least during an appreciable fraction of its existence, as a positive charge surrounded by a negative meson cloud. Thus, when the neutron and electron are separated by a distance less than the extent of the meson distribution (which is of the order of the classical electron radius r ) they will interact electrostatically. The predicted magnitude of this effect depends on the form of the meson theory used in its calculation and, in any event, has only been calcu- lated approximately (i.e., by perturbation techniques, to first order in the coupling constant). Such calculations have been carried out by Slotnick and Heitler (S39), by Case (C2), by Dancoff and Drell (D2), and by Borowitz and Kohn (B48). These calculations predict effects of the order of, but somewhat smaller than, the observed interaction. Thus, for instance, Case (C2) obtained an attractive interaction which, when expressed as a square well potential of range r , yields Vo = 300(/ 2 /2t) ev; f 2 /2w, for this case (the coupling constant for a pseudo- scalar meson with pseudoscalar coupling), is ~5, giving V «* 1500 ev. The second effect arises, as was pointed out by Foldy (F37), as a direct consequence of the anomalous magnetic moment of the neu- tron. It can, in fact, be computed, without specification of the form of the meson theory, 1 by assuming that the neutron obeys the Dirac equation with the additional "Pauli terms" (P7) in the Hamiltonian, — n n (eh/2Mc)(JS(r -H — iPa-E). (n n ^ —1.91 is the neutron moment in nuclear magnetons.) The <r-H term leads to the spin-dependent, mag- netic dipole-dipole interaction between the neutron and the electron. The second term leads to an interaction of the form — yii„(eh 2 /4M 2 c 2 )|8 div E which corresponds, for the field of a point electron, to an attractive interaction with 7o= -(T)(7) 2 (5) 2 " c2 = 4100ev (10) Thus, the Foldy term alone appears almost sufficient to account for the observed interaction. A more accurate measurement of the strength of the neutron-electron interaction should, indeed, eventually lead to an evaluation of the magnitude of the electrostatic interaction and permit a check on the applicability of specific meson theories. 1 Since the fact that the neutron has an anomalous magnetic moment is, itself, a consequence of meson theories, it could not be said that this effect is of non-mesonic origin. Indeed, some of the meson-theoretic calculations (S39, D2) include the effect. However, since no existing theory is capable of yielding the observed value of the neutron magnetic moment, it appears more reasonable to follow the method of Foldy, which assumes the observed neutron moment, and to compute this effect by a phenomenological treatment. 246 The Neutron [Pt. VII The two effects described above are susceptible of relatively simple, order-of-magnitude computations. The strength of the electrostatic neutron-electron interaction has been estimated by Fermi and Marshall (F23), who considered a model of a point proton (charge g 2 e) surrounded by a negative meson cloud of charge density ' w --(^M-ir) <"' (13) This expression is suggested by Yukawa's original (scalar) theory of the meson field, according to which the extent of the meson wave function is essentially determined from the uncertainty principle, a ^ h/KC (12) where k is the pion mass («275 electron masses), g 2 ~ 0.1 to 0.5 is the is the fraction of time during which the neutron exists in the proton- meson state. The potential energy U(r) of interaction of the above charge distribu- tion with a point electron can be obtained by solution of the Poisson equation, and the strength of the neutron-electron interaction is given by b = 4x f U(r)r 2 dr = - (-) g 2 e 2 a 2 Assuming a potential well of radius r and depth Vo, we obtain 36 /g 2 \ /m\ 2 /hc\ 2 , Considering that the computation is classical and, furthermore, modeled on a meson theory which is known to be inadequate, the agreement with the meson-theoretic computations, quoted in the preceding, is quite satisfactory. It is somewhat more difficult to give a simple physical interpretation of the interaction which arises from the neutron's magnetic moment. However, Weisskopf l has suggested a rather ingenious derivation of the Foldy term: The assumption that the neutron obeys the Dirac equa- tion leads to a zitterbewegung in the motion of the neutron, such that its path is a spiral whose radius is of the order of the neutron's Compton wavelength, R ~ h/Mc. The neutron travels with the velocity of light, c, in this spiral path whose pitch is such as to give a transport velocity equal to the velocity v of the neutron. (The intrinsic spin angular momentum of the neutron can be interpreted as arising from this spiral motion.) Accordingly, when the neutron is within a distance 1 Private communication. Sec. 2A1 Interaction with Nuclei 247 R of the electron, which, we assume, behaves like a point charge, there is a magnetic spin orbit interaction between the electron current and the neutron's intrinsic magnetic moment whose energy is, neglecting numerical factors, '"-(s)^-^ (15) Since this interaction has a range of ~R, we obtain -ER 3 /hc\ 2 /m\ 2 , T --z-~-*k)Kz) m * (10s> which is, aside from the factor -f-, the result of the calculation of Foldy. Since the interaction depends only on the electron's charge, it is easily seen to be spin-independent. SECTION 2. INTERACTION WITH NUCLEI A. Introduction The experiments through which the existence of the neutron was established and in which its properties were first elucidated very soon indicated that neutrons are capable of inducing nuclear transmutations. Some of these early investigations have been discussed in Section 1. In these the neutron-induced transmutations were observed in cloud cham- bers. Although the cloud chamber is a very effective instrument for observing nuclear transmutations, its use imposes serious restrictions on the types of nuclear reactions which can possibly be detected: The tar- get nuclei must be capable of being introduced, as an appreciable con- stituent, into the cloud chamber; the transmutation products must be charged and have sufficient energy (range) to leave observable and identifiable tracks. With relatively few exceptions, cloud chamber techniques can only be applied to the study of nuclear transmutations induced in light elements by fast neutrons. With the announcement by Curie and Joliot (C41), in January 1934, of the production of artificially radioactive nuclei, an entirely new field of neutron investigation became available. 1 Their discovery prompted 1 The discovery by Curie and Joliot was made in experiments involving the bom- bardment of boron, magnesium, and aluminum by polonium alpha-particles. They first reported the reaction AF + a. -> P 30 + n p30 _> gi 30 + e + and proved that the reaction product is an isotope of phosphorus, decaying by posi- tron emission, with a half-life of 3.25 min. 248 The Neutron [Pt. VII Fermi to investigate the possibility of producing artificial radioactivity by neutrons, and he soon announced (F13) that the bombardment of aluminum and fluorine by neutrons does indeed induce such radioactiv- ity. Within a short time, Fermi and his co-workers at the University of Rome succeeded in inducing artificial radioactivity in a variety of elements through neutron bombardment (F12). In the next few years the Rome group played a leading role in the investigation of neutron- induced nuclear reactions. A significant step forward in the study of neutron reactions resulted from the observation by Fermi, Amaldi, Pontecorvo, Rasetti, and Segre (F14) that the neutron-induced radioactivity in silver can be sig- nificantly increased by interposing, between the source of fast neutrons and the silver detector, a slab of paraffin. This effect was correctly ascribed to the slowing down of the neutrons by collisions with hydrogen nuclei, and it indicated that the probability of neutron capture increases with decreasing neutron energy. This conjecture was confirmed by the observation, by Moon and Tillman (M35), of an effect of the tempera- ture of the moderator on the neutron capture. The slowing down of neutrons in hydrogen-containing materials will be discussed subse- quently. For the purpose of this summary it is, however, important to note that the recognition of the strong dependence of neutron capture probability on the neutron velocity was a significant step toward the understanding of neutron reactions and that, for many years, sources of neutrons slowed down in paraffin were the main tool for the investi- gation of neutron-induced nuclear reactions. The fact that radioactivity can be induced by neutrons in the heaviest elements is already a strong indication that the reactions responsible for their production probably do not involve the emission of charged parti- cles since, for heavy nuclei, the Gamow barrier is a very strong deterrent to charged particle emission. Furthermore, in many cases the radioac- tive nuclei were shown to be isotopes of the bombarded nuclei (F12). In particular, the ~15 hr half-life /3 — activity resulting from neutron capture by Na 23 , discovered by Bjerge and Westcott (B32), was identi- fied as an isotope of sodium by Amaldi, D Agostino, and Segre (A16) ; the fact that the decay involves emission of p~ particles was interpreted to imply that the radioactive isotope is Na 24 (rather than Na 22 ), indi- cating an (n,y) reaction. The observation that the probability of pro- ducing radioactivities in heavy nuclei increases with decreasing neutron velocity represents an even stronger argument for the contention that the reaction involved is that of radiative capture (n,-y) — capture of a neutron followed by gamma-ray emission. That this is indeed the case was shown by Amaldi, DAgostino, Fermi, Pontecorvo, Rasetti, and Sec. 2 A] Interaction with Nuclei 249 Segre (A19), who proved that the capture of slow neutrons by cobalt, cadmium, chlorine, iridium, silver, and mercury is accompanied by gamma-ray emission. 1 For slow neutrons, radiative capture is usually the most important reaction. However, in the case of the capture of slow neutrons by light nuclei, charged particle emission may be much more probable. Thus, Chadwick and Goldhaber (C9) and, independ- ently, Amaldi, D'Agostino, Fermi, Pontecorvo, Rasetti, arid Segre (A17) showed that the strong capture of slow neutrons by lithium and boron is due to an (n,a) reaction. Somewhat later Burcham and Gold- haber (B75) demonstrated that the capture of slow neutrons by nitrogen is due to an (n,p) reaction. Perhaps the most important advance in the study of neutron reac- tions was the discovery of slow-neutron resonances — the preferential capture of neutrons of specific energies by certain nuclei. The first indi- cation of resonance effects was obtained in the experiments of Bjerge and Westcott (B33), soon followed by the work of Moon and Tillman (M35, T9), which established the resonance nature of some slow-neutron interactions. There then followed a period of rapid development of tech- niques for studying the properties of slow-neutron resonances. Among the most important of these were: the use of cadmium difference and self -indication methods by Fermi and Amaldi (F16) and by Szilard (S64) ; the measurement of neutron slowing down lengths in paraffin using, as a detector, the resonance in question, by Amaldi and Fermi (A20); the use of boron absorption techniques for the measurement of resonance energies by Frisch and Placzek (F45) and by Weeks, Livingston, and Bethe (Wll) ; the development of the first slow-neutron velocity selector by Dunning, Pegram, Fink, Mitchell, and Segre (D20). These tech- niques, their development and extension, will be discussed in detail in Section 3. Among the most significant of the early studies was the proof, by Dunning, Pegram, Fink, and Mitchell (D19), that the very large interaction of thermal neutrons with cadmium cannot be due to neutron scattering and must, therefore, be assumed to result from the radiative capture process. The study of slow-neutron resonance capture received great impetus from the theoretical considerations of Bohr (B44) on the role of the com- pound nucleus in nuclear reactions. On the basis of Bohr's arguments, it became possible to understand the existence of slow-neutron capture resonances, and to interpret their observed properties in terms of the Breit-Wigner formula (B60) (previously derived on the basis of very 1 The emission of gamma-radiation in the capture of neutrons by hydrogen had previously been observed by Lea (L10). 250 The Neutron [Pt. VII general considerations) for the energy dependence of the nuclear cross section in the neighborhood of a resonance. In addition to the work mentioned above, some of the investigations which were important in confirming the theoretical ideas concerning neutron resonances include those of Preiswerk and Halban (P30), Frisch (F46), and Goldsmith and Rasetti (G19). The experiments and theory of slow-neutron resonances, up to 1937, are summarized by Bethe (B24), by Bohr and Kalckar (B45), and by Moon (M36). The theory and observation of slow-neutron resonances, up to 1940, are discussed by Peierls (P10). While the most important advances in neutron physics were made in the study of the properties of slow neutrons, the investigation of fast neutron reactions was not completely neglected. Of the artificial radio- activities induced, by neutrons from Rn-Be sources, in light elements, the pioneering work of Fermi and co-workers (F12) showed, by chemical separation of the radioactive nuclei, that the responsible reactions were mainly (n,p) and (n,a). In the case of the heavy elements, the radio- active products appeared, in most cases, to be isotopes of the target nuclei. Fermi and co-workers advanced two alternative reaction possi- bilities: radiative capture, or the ejection of a neutron from the target nucleus by a neutron-neutron collision, i.e., the (n,2n) reaction. The dependence of the probability of most neutron reactions on the neutron velocity, discussed above, led to the assumption of the first hypothesis, confirmed by the observations on sodium (B32, A16) and other ele- ments (A19). However, in 1936, Heyn (H56) proved that, for neutrons of sufficiently high energy, the (w,2n) reaction could also be induced. In the meanwhile, Dunning and co-workers at Columbia University (D18, D19) were studying the reactions of fast neutrons with nuclei, using Ra-Be sources. In addition, with the discovery, by Szilard and Chalmers (S63), of the photodisintegration of beryllium by the gamma- rays from radium and its products, there became available neutron sources in the intermediate-energy range. These were exploited by a number of investigators (M22, L13, G25) for the study of neutron reac- tions in the 0.1 to 1 Mev range. Of primary importance in the study of fast-neutron reactions was the discovery, in 1933, of the possibility of producing strong fast-neutron sources by the use of particle accelera- tors through (d,n) reactions (Crane, Lauritsen, and Soltan, C30), and (p,n) reactions (Crane and Lauritsen, C31). Of special significance in this respect is the d + d reaction (Oliphant, Harteck, and Rutherford, ()3), which can be used with relatively low-voltage accelerators. Among the early results of fast-neutron studies was the observation by Lea (Lll) of the excitation of gamma-rays by the passage of fast Sec. 2B] Interaction with Nuclei 251 neutrons through matter. Lea showed that the production of gamma- rays increased with the atomic number of the material traversed, and that the observed effects were not due to a neutron capture process; he ascribed them to the excitation of nuclei in the inelastic scattering of fast neutrons, followed by nuclear de-excitation through gamma-ray emission. Similar results were obtained by Kikuchi, Aoki, and Husimi (K12). It was shown, by Danysz, Rotblat, Wertenstein, and Zyw (D3), by Ehrenberg (E2), and by Collie and Griffiths (C20), that fast neutrons are indeed slowed down, far in excess of what could be ac- counted for by elastic scattering, in their passage through heavy ele- ments. The last of the significant new neutron reactions to be uncovered was neutron fission, discovered by Hahn and Strassman (H4) early in 1939. This discovery was the result of a long series of investigations which attempted to understand and interpret the results of the neutron bombardment of uranium, first reported by Fermi et al. (F12). Imme- diately after the announcement by Hahn and Strassman that isotopes of barium, lanthanum, and cerium are certainly among the products of the capture of slow neutrons by uranium, Meitner and Frisch (M23) showed that a fission reaction (division into two fragments of roughly equal mass) was to be expected from energetic considerations; such a reaction, they observed, is highly exoergic. Physical evidence for the expulsion of high-speed nuclear fragments in the slow-neutron fission of uranium was soon obtained by Frisch (F49) and by Joliot (Jll). The history of the discovery and early investigation of the fission reaction has been reviewed by Turner (T17). B. General Considerations Given a complete knowledge of the law of force between nucleons and of the law of motion (quantum mechanics) which governs their mutual interaction, it is conceivably possible to set up a program of computing the static properties of all nuclei and the detailed characteristics of the interactions between neutrons and nuclei. Actually, this program is still in its preliminary stages, if indeed the goal is at all attainable for any but the lightest nuclei. In the first place, the law or laws of force are not yet sufficiently well established. In the second place, their appli- cation to complex, heavy nuclei would involve the solution of the equa- tions of motion for the many-body problem, in which the components are closely spaced and strongly interacting (so that the approximation methods, which are so useful in problems involving the electrons in an atom, may not be applicable to nuclei). Finally, it is by no means cer- tain that specifically many-body forces (not detectable in or predictable 252 The Neutron [Pt. VII from a study of the nucleon-nucleon interaction) do not come into play in the interaction of the nucleons in a nucleus. Nevertheless, a number of attempts have been made to understand the properties of nuclear (especially neutron) interactions in terms of approximations in which the nucleons in the nucleus are assumed to behave as more-or-less independent particles (A19, Bll, B21, Pll). The predictions of such models were soon found to be at strong variance with experimental observations, especially with regard to the properties of slow-neutron resonances; for, while these models predict strong ther- mal neutron capture cross sections and slow-neutron resonances, the widths of the levels and the spacing of levels, as predicted, were far in excess of those observed. Furthermore — and most significant — the independent particle models were unable to account for the predomi- nance of radiative capture over scattering in most slow-neutron reso- nances. To overcome these difficulties Breit and Wigner (B60) introduced the idea that the slow-neutron capture process must involve more than one of the nuclear constituents; their arguments were based mainly on an analogy with certain atomic and molecular phenomena. The conceptual and theoretical groundwork for the understanding of nuclear reactions was laid by Bohr (B44) in a work of classic and far-reaching significance. Bohr emphasized the necessity of going to the opposite extreme from the independent particle picture, and of recognizing that the nucleons in the nucleus interact very strongly with each other. Thus, a neutron, on entering the nucleus, very rapidly loses its identity as bombarding particle, sharing its energy among all the constituents of the nucleus. Accordingly, a nuclear reaction must be regarded as taking place in two distinct and separable stages. In the first, the incident particle is in- corporated into a compound nucleus, sharing its energy — kinetic and binding — with the rest of the nucleus. In the second stage, the com- pound nucleus gives up its excitation energy by any one of the possible means at its disposal. These means include radiation, particle emission, and neutron re-emission. In this competition among the various (ener- getically) possible de-excitation processes, radiation can compete very favorably with particle emission, especially in heavy nuclei, since parti- cle emission (including neutron re-emission) requires the concentration of a large fraction of the excitation energy into one particle, a process which is relatively improbable. Furthermore, because of the sharing of the excitation energy among the nuclear constituents, the compound nucleus has a relatively long lifetime, which results in comparatively sharp resonances. These ideas lend themselves to quantitative de- velopment in terms of a statistical theory of nuclear energy levels and Sec. 2B] Interaction with Nuclei 253 of nuclear reactions (B45, B23, F43, W16, L2) ; they provide the basis for many theoretical investigations of the properties of slow-neutron resonances (B25, K2, S37, B61, W29, F27, F28, A6). In their extreme form, they suggest the liquid drop model of heavy nuclei, which has been so useful in understanding the dynamics of the fission process (B46). So successful were the ideas of Bohr, and their quantitative develop- ment, in understanding, correlating, and predicting the properties of nuclear reactions, as well as some of the general features of the stable nuclei (binding energies, curve of stability) that the independent parti- cle model was completely discredited. There remained, however, cer- tain features of nuclear behavior which, although they could be fitted into the general framework of the statistical model, required concessions in the direction of the independent particle model; thus, it was neces- sary to treat somewhat differently nuclei with odd and with even num- bers of nucleons. Furthermore, the success of the independent particle model, as applied by Schmidt (S5), in correlating nuclear magnetic moments indicated that the ground states, at least, of nuclei require such a treatment. The accumulation of nuclear data has recently led to the recognition by Mayer (M7) that many nuclear phenomena indicate the existence of a nuclear shell structure, first predicted by Elsasser (E4), thereby re- viving the independent particle model. The particular stability of nuclei containing 2, 8, 20, 28, 50, 82, or 126 neutrons or protons is borne out by a large number of nuclear data. A number of forms of the inde- pendent particle model have been invoked to derive the above "magic numbers" (F6, N2), of which the most successful has involved the assumption of a strong spin-orbit coupling, by Haxel, Jensen, and Suess (H49) and by Mayer (M8). Table 3 (after Mayer, M9) partially summarizes the predictions of this model. While the statistical model and the independent particle model repre- sent opposite extremes, it appears necessary nevertheless to recognize that both points of view are applicable, each in its domain, and that the two approaches are complementary. Thus, the independent particle picture appears to be required for the understanding of the ground and low-lying excited states of nuclei, whereas the statistical model is more appropriate for phenomena involving relatively large nuclear excitation, as is the case in almost all neutron reactions. Weisskopf (W17) has pointed out that the nuclear situation seems analogous to that of the electrons in a metal, where the electrons can be described in terms of an independent particle model for the states of small excitation because of the effects of the Pauli principle, despite the fact that they are inter- acting very strongly with each other. 254 The Neutron TABLE 3 [Pt. VII Order of Energy Levels Obtained from Those of a Square Well Potential by Spin-Orbit Coupling (After Mayer, M9; Oscillator Square Well Level Spin Number Number Total Number Term of States in Shell Number Is lSi A 2 2 2 1 lp ipy* 4 lPH 2 6 8 2 Id ldtft 6 (14) 2s &X 2 ld% 4 12 20 3 1/ V» 8 8 28 2p 2PK 4 6 2PK 2 (12) (40) 4 1(7 ig% 10 22 50 2d 2d H ^g% 2d H 3sy 2 6 8 4 2 5 Ih lhi$i 12 32 82 2/ 2/54 8 3p 3pk 3ph 4 6 2 10 6 3d 4s 2?« 14 44 126 Sec. 2B] Interaction with Nuclei 255 Nevertheless, it is necessary to observe that neither model provides a complete description of nuclear phenomena in any energy region; the strong interaction undoubtedly influences the properties of the low- lying nuclear levels, while (as we shall have many occasions to observe) the shell structure has important effects on the characteristics of neutron reactions, even at relatively high levels of excitation. The properties of nuclear reactions have been discussed in Part VI. In this section we shall review these properties as they apply to neutron reactions, and summarize the status of the present knowledge of the interactions of neutrons with nuclei. The available data on neutron cross sections have been compiled by Goldsmith, Ibser, and Feld (G20) (October 1947), by Adair (A2) (July 1950), and by an AEC Committee (A0) (1952). The discussion here will lean heavily on these compila- tions, in which references are given to the original investigations. 1. Energetics of Neutron Reactions. When a neutron of kinetic energy E is captured by the target nucleus, the product (compound) nucleus acquires an excitation of energy E' = E + e. e is the binding energy of the neutron in the product nucleus. This excitation makes possible a variety of nuclear reactions, depending on the particle or par- ticles whose emission from the compound nucleus requires less than the energy E'. The characteristics of the reactions observed depend on the properties of the compound nucleus at the excitation energy E', and on the competition between the possible modes of de-excitation of the compound nucleus. Clearly, for neutrons of kinetic energy E <SC e, the characteristics of the observed reactions will be critically dependent on the value of e. The variation of e with Z and A may be summarized as follows : For nuclei of A < 20, e exhibits large (periodic) fluctuations from nucleus to nucleus. For nuclei of A "> 20, the values of e, on the average, in- crease slowly from -~8 Mev to ~8.5 Mev at A — 130, and then decrease slowly to <~7.5 Mev for the heaviest nuclei. However, these values apply to the stable nuclei; the capture of a neutron by a stable nucleus usually results in a radioactive product nucleus for which the neutron binding energies are somewhat smaller than for the stable nuclei. Thus, the values of e, for captured neutrons, average ~7-8 Mev for A be- tween 20 and 150, falling slowly to ~6 Mev for neutrons captured by the heaviest nuclei. However, neutron binding energies fluctuate significantly from nucleus to nucleus. The capture of a neutron by a nucleus with an odd number of neutrons results in a greater (by ~l-2 Mev) release of energy than capture by adjacent nuclei of even neutron number. Quite large devia- tions from the average may also occur in the vicinity of neutron magic 256 The Neutron [Pt. VII numbers; thus, the value of e for a neutron captured in a nucleus lack- ing but one neutron for a closed shell will be anomalously high, while « for a neutron captured by a nucleus already having a completed neutron shell is anomalously low. Furthermore, such magic number effects appear to persist for neutron numbers ranging considerably to either side of a magic number (K23). In those (relatively rare) cases in which the masses of the target and product nuclei are known, the neutron binding energy can be computed : e = 931.16 {M(X A ) + M(n) - M(X A+1 )} in Mev (16) For light nuclei the masses of the nuclei involved are, in general, avail" able and the above computation can be made. For nuclei of A "> 30 the masses are usually not known, and it is then necessary to resort to the general considerations described above, if it is desired to estimate the value of «. From the above expression for e, it may be seen that it is not necessary to know the absolute value of the nuclear masses in order to compute e, but only the mass difference, M(X A+1 ) — M(X A ). Nuclear mass dif- ferences are frequently known as a result of measurements of the energies of reactions involving the two nuclei in question, even when the absolute values of the masses are unknown. Thus, in the region of the naturally radioactive nuclei, neutron binding energies can be computed with the aid of the observed energies of alpha- and beta-emission (W9). For a few nuclei the energy release accompanying slow-neutron cap- ture has been measured directly by observation of the energy of the gamma-radiation emitted when the excited compound nucleus decays to its ground state. In this fashion, Bell and Elliott measured the binding energy of the deuteron (B16). Unfortunately, for nuclei more compli- cated than the deuteron the radiative de-excitation of the compound nucleus is usually achieved by the emission, in cascade, of a number of gamma-rays. Since there are usually a large number of energy levels of the compound nucleus available to such cascade gamma-ray emis- sion, the capture gamma-ray spectrum is quite complicated. However, in certain favorable cases the emission of a single gamma-ray, carrying away all the excitation energy, occurs in a reasonably large fraction of the decays; the energy of this gamma-ray gives, directly, the value of e. Such direct transitions to the nuclear ground states have been observed in a number of nuclei by Kinsey, Bartholomew, and Walker (K15). In general, the nuclear reactions in which we are interested can be represented symbolically as on 1 + Z X A -> ( Z X A+1 )* -* z x a + z _ z Y A+l - a + Q (17) Sec. 2B] Interaction with Nuclei 257 [For many light nucleus reactions the intermediate (compound) nucleus step should be omitted, since it has no independent existence.] The re- action is frequently abbreviated as X. A (n,x)Y A+1 ~ a , or simply as ~K. A (n,x). The Q value of the reaction is determined by the masses of the nuclei involved : Q = 931.16{AT(n) + M(X) - M(x) - M(Y)} in Mev (18) or, conversely, a measurement of the Q value can be used to determine one of the masses, if the other three are known, or the mass difference M (X) — M (Y), from the masses of the neutron and the ejected particle. A positive Q value indicates an exoergic reaction, i.e., a reaction that can take place (from the point of view of available energy) with neutrons of zero kinetic energy; a negative Q value means that the reaction is endoergic and is not possible for neutrons of kinetic energy less than a certain threshold value: f[M(X) + M(n)]) (04 + 1)1 The factor in brackets arises from the conservation of momentum, which requires that some of the neutron kinetic energy be expended in pro- viding for the center-of-mass motion of the system. The possibility that a given reaction can be initiated by thermal neutrons depends on having Q > 0. Two reactions are, however, always possible for thermal neutrons: (1) elastic scattering (or re-emission) of a neutron (n,n), for which Q = 0; (2) radiative capture (n,y), for which Q = e. 2. Theory of Neutron Reactions ; the Compound Nucleus. According to the Bohr model, most neutron reactions can be regarded as taking place in two distinct stages — the formation of an excited compound nu- cleus, and its subsequent decay. In our survey of the theoretical conse- quences of these ideas, we lean heavily on the schematic theory of nuclear reactions developed by Weisskopf and co-workers (F27, F28, W17). This theory is based on some general assumptions regarding the struc- ture of nuclei: that the nucleus has a well-defined surface, which is a sphere of radius R, outside of which the interaction with neutrons is negligible; that, once inside the nucleus, the neutron interacts very strongly with the nuclear constituents, rapidly sharing its excitation energy among them. Because of the saturation property of nuclear forces, the average distance between nuclear constituents is independent of their number, 258 The Neutron [Pt. VII A. Thus, the volume V of a nucleus is proportional to A, and its radius can be approximated by the relationship R = r A 1A (20) with r between 1.3 and 1.5 X 10~ 13 cm. A neutron of kinetic energy E will, on penetrating the nuclear sur- face, suddenly find itself in a region where its kinetic energy is E' ^ E + e, where e is the average kinetic energy of a nucleon inside the nu- cleus. Such a sudden change of kinetic energy results, from wave- mechanical considerations, in a reflection; the probability for penetra- tion of the surface is given by 4JcK P = -tt-t^-, (2D where (k + KY „ 2ME 1 k 2 = — — = - (22a) X 2 for the incident neutron, and K2 = ™<l+A = k2 + K j ,., w. fi 2 h 2 for the neutron inside the nucleus. (K and k are the wave numbers associated with the neutron inside and outside the nucleus, respectively.) The value of K can be computed on the assumption that the nucleons in the nucleus obey Fermi-Dirac statistics, distributing themselves among the various possible states of momentum, spin, and isotopic spin (charge) so that no state is occupied by more than one nucleon, giving £ = ( ) ( — ) ^ 20 Mev V 2M I \16ttVJ (23) K ^ 1.0 X 10 13 cm -1 for r = 1.5 X 10 -13 cm. For slow neutrons, k « K ^ K , and 4fc P ^ — « 1 (24) Ko Despite the smallness of P for slow neutrons, the probability of neutron capture to form a compound nucleus may be quite large since slow neutrons have a large extent (wavelength). The capture probability depends critically on the proximity of the excitation energy (E' = E + e) to one of the energy levels of the compound nucleus, being essentially a resonance phenomenon. We postpone the discussion of Sec. 2B] Interaction with Nuclei 259 the formation of the compound nucleus to the next section and confine ourselves, at this point, to consideration of what happens after the com- pound nucleus has been formed. Once a neutron has entered the target nucleus, it very rapidly shares its energy among the nuclear constituents, forming a compound nucleus in an excited state. Furthermore, owing to the smallness of P (which works both ways), the neutron has a very small probability of re-emerg- ing, even if it should find itself in the relatively improbable circumstance of possessing all the excitation energy. Consequently, the lifetime of the compound nucleus is very long compared to the time required for its traversal by a nucleon. In fact, for slow neutrons captured in nuclei containing a relatively large number of constituents (say A >^ 100) the most favorable mechanism for de-excitation of the compound nucleus is usually the emission of gamma-radiation, which is itself a relatively improbable process. These considerations can be made somewhat more quantitative. The compound nucleus, formed in an excited state, may have a number of (energetically) possible modes of decay which always include radiation (n,y), neutron re-emission (n,ri), and may also include the emission of one or more particles (n,a), (n,b), etc. These compete freely with each other, each mode of decay being characterized by a mean-life, t,- (the lifetime which the excited state would have if all other possible modes of decay were turned off). The mean-life of the excited state 1 ^ 1 -=£- (25) T ,■ Ti is associated, according to the uncertainty principle, with a finite width (energy spread) of the excited state of the compound nucleus: h r = - (26) T Correspondingly, we can define a partial width for the z'th mode of decay: h r,- = - (27) so that r = Er f (28) i The characteristics of the compound nucleus decay are determined by the relative values of the partial widths for the various possible modes of decay. For the case of gamma-ray emission, the entire excitation 260 The Neutron [Pt. VII energy is available to the gamma-radiation. Since, for slow neutrons, e » E and E' « e , the probability of gamma-ray emission is essentially independent of the energy of the incident neutron. However, T y « constant may be expected to depend to a certain extent on the angular momentum properties of the state involved, so that the constant will not be the same for all the states of the compound nucleus. For particle emission, however, the situation is quite different. In this case, in addition to kinetic energy, the emerging particle must also be supplied with its binding energy. Since the excitation energy is shared among the many particles in the compound nucleus, the possi- bility of emission of a particle depends on the concentration of sufficient energy into this particle to allow it to escape. Let T a be the average time between such rearrangements of the nuclear constituents as would permit the emission of particle a. The frequency of emission of the particle a (l/r a ) is then given by the product of the frequency of such favorable configurations (1/T a ) and the probability that the particle a, given the requisite amount of energy, can penetrate through the nuclear surface : 1 Pa - = V (29) Ta i-a Hence h hP„ r a = - = — (30) T a ±a According to the previous considerations, and- assuming that the kinetic energy of the emerging particle is small compared to its energy in the nucleus, and that it has zero angular momentum, 4fc„ P a = — Qa (31) -fro where G a is the Gamow barrier for particle a; G n — 1 for neutrons. Thus 4hG r„ ^ — — K = 7a G a E a * (32) The above relationship applies only to the emission of particles of zero angular momentum. It involves the kinetic energy of the emerg- ing particle, E a , through the factor E a y * and through the energy de- pendence of the Gamow factor, (?„, when the particle is charged. It involves the properties of the compound nucleus through the factor K 0) which is essentially independent of the nucleus involved, and the perio- dicity T a . The last factor depends quite critically on the specific nu- Sec. 2B] Interaction with Nuclei 261 cleus involved and on the excitation energy of the compound nucleus state; this dependence on the properties of the compound nucleus is through the nuclear level spacing, at the excitation energy involved, as indicated by the following simplified considerations, due to Weisskopf (W17): Assume that the energy levels of the compound nucleus are equally spaced, E n = E + nD (33) Owing to the strong interaction, the nuclear wave function is given by a linear combination: £, -iE n t -iE t * -inDt * = 2-, a n<l>n exp — - — = exp — - — 2-, a n<t>n exp — - — (34) n=0 U B- n=0 h The periodicity T, which is defined as the time required for the nucleus to return to a previous configuration, is given by the condition | *(t + 2xh/D) | = | ¥(t + T) | = | ¥(*) | (35) whence 2irh T = — (36) Although the level spacing in the compound nucleus is far from uni- form, the above relationship nevertheless holds approximately for our case, where D a is the average level spacing (of the levels which are capable of decaying through emission of the particle a in a state of given angular momentum) in the energy region of the excitation energy. Combining Eqs. (32) and (36), 2G a D a r« « — k a = CD a G a E a 1A (37) for the emission of particles of zero angular momentum. The above is a much simplified derivation of a well-known relationship between level spacing and level width (B21, B46, F27), which is borne out by the observed neutron widths for slow neutrons (W31, B34). The E y% dependence of the particle width on its kinetic energy also follows from very general considerations involving the density of mo- mentum states available to the outgoing particle, in the case of emis- sion of zero angular momentum particles. For the emission of particles of higher angular momentum, I, there is an additional energy depend- ence, which can be expressed as follows: r„; = (21 + l)CD al G a iE a ^v al (38) 262 The Neutron [Pt. VII where v a i = vi(k a R), R being the nuclear radius. Expressions for vi{x) are (F27, B34) f o = 1 (39) x 2 v l (1 + x 2 ) X 4 V% (9 + 3x 2 + * 4 ) v 3 = (225 + 45z 2 + 6x 4 + a; 6 ) These have the limiting values y2l vi — > 1 for x — > oo Thus, for the emission of low-energy particles (k a R « 1), we need con- cern ourselves only with I = 0. At sufficiently higher energies, states of i! > come into play. Because of the selection rules associated with angular momentum changes, the absorption and emission of particles of different angular momenta usually involve different energy levels of the compound nu- cleus. For a given energy level of the compound nucleus, the emission of a given particle in different states of angular momentum can usually be treated on the same basis as the competition for the emission of dif- ferent particles. So far, the spacing of the levels in the compound nucleus, D, has en- tered our considerations only as a parameter in the level width formulas. Clearly, the level spacing is also of primary importance in the formation of the compound nucleus, since the probability of neutron absorption depends on the proximity of the nuclear excitation energy to one or more of the levels of the compound nucleus. On the basis of the Bohr model, it is possible to obtain a rough, semi- quantitative picture of the dependence of the level spacing of the nu- cleus on the excitation energy. For low excitation energies (of a few Mev or less) the possible modes of excitation of the compound nucleus are few, and the energy levels are widely spaced. The excitation of the low-lying levels will involve only one or a few nucleons, and the details of the low-lying levels will depend on the nucleus involved, being Sec. 2B] Interaction with Nuclei 263 strongly influenced by such phenomena as magic numbers. As the en- ergy of excitation is increased, it becomes possible to excite more modes, involving a greater number of nuclear constituents, and the levels be- come more closely spaced. For large excitation energies (say ~10 Mev) an expression for the spacing of energy levels may be obtained from thermodynamic considerations, and is of the form w(E') = Cexp2(aE') 1A (41) w(E') = 1/D is the level density (number of levels per unit energy inter- val) at the excitation energy E'. Typical values of C and a, for a number of odd values of A, are given in Table 4; the level densities for even A nuclei appear to be somewhat lower. TABLE 4 Constants Determining the Level Spacings of Some Odd A Nuclei, Eq. (41) (Derived by Blatt and Weisskopf, B34, from available data) A C (Mev" 1 ) a (Mev" 1 ) 27 0.5 0.45 55 0.33 1.3 115 0.02 6.5 201 0.005 12 Figure 4a is a schematic energy level diagram for two nuclei, A ~ 55 and A = 201. Aside from the details, which depend on the nucleus in- volved, the general characteristics of the level scheme for the heavier nucleus can be derived from that of the lighter by a contraction of the energy scale. This feature is qualitatively expected from the Bohr model since the heavier the nucleus, the more particles are available to participate in the excitation, and the greater the level density at a given excitation energy. Figure 4b shows the same level diagrams greatly magnified in the region of excitation corresponding to the absorption of a neutron. As implied by Fig. 4b, the level spacing of interest, in considering reactions involving neutron capture, is that which corresponds to the excitation E' — E + « of the compound nucleus. At first glance it might be expected that, for neutrons of moderate energy, this would be primarily determined by the binding energy, e. However, evidence has recently been presented by Harris, Muehlhause, and Thomas (H40) and by Hurwitz and Bethe (H81) that this is not the case. Thus, Harris et al. have presented data which indicate that the level densities at E' 264 The Neutron [Pt. VII 10 -P 5 AZ55 e~ A = 201 Fig. 4a. Schematic energy level diagrams for two nuclei, A =* 55 and A « 201. The average level spacings are according to the statistical theory (Eq. 41), and the constants are from Table 4. The details of the structure, i.e., the relative positions of adjacent levels and the fluctuations in level spacing, are entirely fictitious, cor- responding to no known nucleus. They are intended to illustrate the possibility of large deviations from the average. Sec. 2B] Interaction with Nuclei 265 « e, for capture of neutrons by odd proton-even neutron and by even proton-odd neutron nuclei, are essentially the same (for nuclei of roughly the same A) although e is considerably smaller for the former than for Mev 1.0 0.9 0.8 0.7 0.6 6q II &1 0.5 0.4 0.3 0.2 0.1 A ==55 AS 201 Magnification X 10 X 20,000 500 400 300 200 100 Fig. 4b. Energy level diagrams of Fig. 4a greatly magnified in the region of the neutron binding energy. the latter (by ~1 Mev). They ascribe the difference in e to a lowering of the ground-state energy of the compound nucleus of even Z-even N, as compared to that of odd Z-odd N; the level densities corresponding to e are, however, the same for both types. Hurwitz and Bethe concur and 266 The Neutron [Pt. VII present arguments for the hypothesis that the level density of the com- pound nucleus, at E' « e, is determined primarily by the binding energy of the target nucleus (rather than by that of the compound nucleus); target nuclei with high binding energy (e.g., even-even or magic number nuclei) result in compound nuclei with low level spacing. Data of Hughes and co-workers * on the radiative capture of fast neutrons, how- ever, appear to be in disagreement with this point of view. In any event, it should be pointed out that the effect of the apparent validity of the independent particle model at low excitation energies will be to invalidate the applicability of Eq. (41) to excitation energies of a few Mev. Available evidence, mainly from inelastic scattering and (p,n) reactions, indicates that level densities at low excitation energies are rather smaller than predicted by Eq. (41), and that the exponential be- havior does not really set in until values of E' >, 3-5 Mev are achieved. The constants of Table 4 should be regarded as applying, roughly, to energies E = E' — a ~ 0-2 Mev. The compound nucleus theory of nuclear reactions, outlined above, will be expected to break down at very high excitation, for at high enough energies the probability of escape of the incident neutron after a single traversal may be quite large, i.e., the nucleus is more or less transparent to high-energy neutrons. Thus, the incident neutron may not have sufficient opportunity to share its energy among the nuclear constituents before it (or one of the other nucleons) escapes. In this respect, it is to be expected that the absorption of a charged particle by a heavy nucleus will, even for high incident energies, be more likely to lead to a true compound nucleus, since the escape of the charged particle from the nucleus is impeded by the Gamow barrier. 3. Formation of the Compound Nucleus; Resonances and the Breit- Wigner Formula. In accordance with the two-stage picture of neutron reactions (formation of a compound nucleus; subsequent emission of a), the (n,a) cross section is <j{n,a) = <r c — (42) The first factor, <r c , is the cross section for the absorption of the incident neutron into an excited compound nucleus; the second factor represents the relative probability for the de-excitation of the compound nucleus through the emission of a. The cross section for the formation of the compound nucleus depends critically on the proximity of the excitation energy (E + e) to one or 1 Private communication. Sec. 2B] Interaction with Nuclei 267 more of the energy levels of the compound nucleus. In the case where only one level of the compound nucleus is involved (all other levels being far enough removed from the value of the excitation energy so that their influence on a c may be neglected), the energy dependence of <r c is given by a characteristic resonance formula, r„r <r c = ttK*1 — ■ (E - i whence <r c = tX 2 / ~-^r — (43) J (E - E T f + r 2 /4 a{n,a) = irX 2 / ~ — (44) K (E - E r ) 2 + T 2 /4 Equation (44) is the well-known Breit-Wigner one-level formula; the symbols have been previously defined, except for E T , which is the neu- tron kinetic energy for which the excitation energy is equal to the dif- ference between the ground state and the energy of the compound nu- cleus level involved in the reaction, and /, a statistical weighting fac- tor which depends on the neutron spin, s = \ ; its orbital angular mo- mentum, lh; the total neutron angular momentum quantum number j = I ± \ ; the spin of the target nucleus, J; and the spin of the com- pound nucleus level involved, J = I+j,I + j— 1, • • -, \ I — j\: (2/+1) _p/ + l) (27+l)(2s+l) 2(27+1) The factor / is simply a measure of the probability that the neutron angu- lar momentum, jh, and that of the target nucleus, lh, either or both assumed oriented at random (unpolarized), will have a relative orienta- tion appropriate to the necessary total angular momentum, Jh, required for the formation of the compound nucleus level involved in the reso- nance. For slow neutrons, I = 0, j = \, and J = I ±\; whence for I ± 0, 1 1 1 ± 1 f^/ . for large I (45a) 2L (27+l)J 2 and /o = 1 for 7 = For neutrons of I > there are two possible "channel" spins, j = I ± |, each of which combines with the target nucleus spin 7 to give the possible compound nucleus spin values J. Some of these can be formed through only one channel; for these, the statistical factor, fi(J), is un- ambiguously given by Eq. (45). Other J values can be formed through both channels; for such levels the question arises as to whether the proper value of the statistical factor is fo(J) or 2f t (J). 268 The Neutron [Pt. VII Let us consider a specific example: Assuming a P-resonance (1=1) for a target nucleus of / = §, the possible J values of a compound nu- cleus state are J = 2,1 fori = f J = 1,0 forj=f The statistical weights are, from Eq. (45), /i(2) = f /i(l) = f /i(0) = i For levels of J = 2 or J = 0, there is no ambiguity. The J = 1 state illustrates the possible ambiguity. On the one hand, a given J = 1 level may be such that it interacts only with one of the two possible channel spins (or one of two possible orthogonal linear combinations of / = -J and j = |); under these circumstances, the statistical factor /i(l) = i- O n the other hand, the interaction might be completely independent of j, depending only on the I value, which is the same for both channels. In this case, /i(l) = 2 X f = f • In either event, the ambiguity disappears if many levels of the com- pound nucleus are involved in a given interaction, since the sum of the statistical factors, over all possible J and j values, is ^2fi(J) = 3. This is a special instance of the general relationship J,i T,fi(J) = (2l+l) (45b) The situation with regard to the statistical factor is even more com- plicated if the same compound nucleus state can be formed by different neutron orbital angular momenta (12). However, this added compli- cation is seldom of practical importance, since the neutron widths will be very different for different values of I, which fact usually effectively eliminates the influence of all but the lowest possible I value. In addi- tion, for a given compound nucleus state the parities of both the initial and the final states are well defined. Hence, the choice of possible I values is even further limited, adjacent values not being simultaneously available. According to Eqs. (43) and (44), neutron capture and the accompany- ing reactions are characterized by sharp resonances for incident neutron Sec. 2B] Interaction with Nuclei 269 energies E = E r , with the maximum (peak) cross section 47rX 2 / z r ni cole - r (46) 4irX 2 fiT n iT a <r i(n,a) = (47) and (full) width at half maximum h ae = r = - T (48) Equation (44) applies to the case of elastic scattering as well as to the case of emission of some other particle. However, in addition to the scattering which involves the formation of a compound nucleus (so- called capture scattering) there is a second type of elastic scattering process known as potential scattering. Potential scattering results from the small penetrability of the nucleus to slow neutrons (Eq. 24) and can usually be described in terms of the scattering of an impenetrable sphere, for which the scattering cross section would be op = ttX 2 £ (21 + 1)| e** - 1 | 2 = 4*-X 2 £ (21 + 1) sin 2 & (49) i i |j is the phase shift of the scattered partial wave of orbital angular momentum Zh. For slow incident neutrons, R <5C X = 1/fc and So = kR & >0 « So (50) whence <r p ^ a p0 = 4xX 2 sin 2 kR ^ 4irR 2 (49a) The phase shifts corresponding to scattering by a rigid sphere can be obtained from the following general expression (F27, L23), with kR = x where J and N are, respectively, the Bessel and Neumann functions. So = x x _ Si = x h cot x _. x 2 - 3 £ 2 = x — x + cot ■ Sx 3x x(x 2 - 15) £q = a: H cot — ?3 2 fa 8 - 15 (50b) 270 The Neutron [Pt. VII For x « 1, So = x x 3 x 5 x 7 /x b x 1 \ \7 21/ ) (50c) * 3 ~225\7 ~ 7' A more useful approximation is, in the limit x <JC | 21 — 1 |, x 2l+l & « tan -1 (50d) At the other extreme, for x 5S> I 2 , It 6 « a: - - (60e) Values of the phase shifts vs. x have been tabulated by Lowan, Morse, Feshbach, and Lax (L23). Equation (49) applies to the general case of scattering by an arbitrary potential (as well as to scattering by a rigid sphere) : <r,c = 4ttX 2 J2 W + 1) sm 2 ft (49') i The phase shifts, ft, are obtained from the asymptotic behavior of the radial functions, u h in terms of which the wave function of the system is written : ii>(r,0) = 22 Ui(r)Pi(cos 0) (49") i The functions ui satisfy the radial Schrodinger equations: «i" + ;?>— ^ ui = (49"a) For slow and intermediate neutrons, X ^> R and only I = scattering need be considered; i.e., ft> <3C /3 , and c sc = 4xX 2 sin 2 /3 . The value of ft, depends, of course, on the form of the scattering po- tential V (in Eq. 49"a, with 1 = 0). A number of possibilities are illus- trated in Fig. 5 in which u is plotted in a number of hypothetical cases. Sec. 2B] Interaction with Nuclei u (r) 271 Fig. 5. Illustrative examples of the joining of the neutron wave functions at the nuclear boundary, leading to different phase shifts and scattering lengths. Case (a) applies far from a resonance, on the assumption of a "rigid sphere" nuclear model. Cases (b), (c), and (d) represent resonance scattering. Case (e) is intended to illustrate scattering by a finite potential well. 272 The Neutron [Pt. VII It is assumed that the potential V acts only in the range — R; foj r > R, u (r > R) = sin (kr + p ). The intercept a of u (r > R) on th< r axis, indicated with an arrow in each case, determines /3 = — a/%. a is called the scattering length or scattering amplitude; according tc convention, positive a will be assumed to mean that the intercept is on the right — cases (a), (d), and (e) — i.e., a negative phase shift, /Jo- Figure 5a illustrates the case of scattering far from a resonance; it is seen that a ^ R, /3 ^ -R/X « 1, and <r sc ^ AtvR 2 . (Figure 5 greatly exaggerates the magnitude of R; for slow neutrons, R, the nuclear radius, is ~10~ 12 cm and X is ~10 -8 cm.) This case is essentially the same as scattering by a rigid sphere, for which u (r < R) = 0. Cases (b), (c), and (d) represent various stages of scattering near and at resonance. It is seen that the resonance condition (case c) corre- sponds to a ^ ±X/4, ft, ^ ±x/2, and <r ^ 4ttX 2 . Case (e) is supposed to represent the class of scatterings by a finite potential well, rather than by a rigid sphere (e.g., neutron-proton scat- tering). Under such circumstances, the scattering length depends in detail on the depth and range of the potential. In particular, if we con- sider scattering by a potential of fixed depth but continuously increasing range, the value of a will go through "resonances" (involving both large magnitude and change in sign) at certain values of R. Such "nuclear size resonances" have recently been invoked by a number of authors (F38, M40, S33) to explain large variations in the potential scattering of adjacent nuclei. For neutron energies far from a resonance, potential scattering is the most important scattering process. In the region of a resonance, cap- ture scattering predominates. However, for the scattering of a neutron of given I, the capture and potential scattering are coherent and inter- fere with each other. Thus, the Breit-Wigner formula must, in the case of elastic scattering, be modified to a(n,n) = 7rX 2 /j i?ni e 2i c — 1 2 + <r/ (51) (E - E r ) + iT/2 where <r p ' is the incoherent part of the potential scattering, 1 a p ' = a p - tX% e 2iit - 1 | 2 (49b) For most observed neutron resonances, the neutron kinetic energy is sufficiently low so that we have only to concern ourselves with I = interactions. Furthermore, except for the light nuclei (A ^ 25), the 1 Note that the usual coefficient (21 + 1) is contained in the statistical factor /;. Sec. 2B] Interaction with Nuclei 273 only reactions of significance in the region of observed resonances are (n,ri) and (n,y). For such resonances (with kR <<C 1), r = r» + r r <r(n,y) = 7rX 2 / - -L n* y (E- - E r ) 2 + T 2 /4: <r(n,ri) = irX 2 fo r„ (E - E r ) + tT/2 (52) + 2kR + 4wR 2 (l - /„) The radiative capture resonance is of the symmetrical Breit-Wigner form (Eqs. 43 and 44). Because of the interference between capture and potential scattering, the scattering resonance is not symmetrical about E r . Instead, it goes through a minimum at an energy below E r and falls off less rapidly on the high-energy side. The position of the scattering minimum is given by r, r„x D *'-**- a* -WW* (53) and the cross section at the minimum is <rmin(n,n) ^ 4xE 2 ( 1 - /o + -^— j- ) (54) (assuming T n X/R <<C T and using Eqs. 37 and 52). Figures 6a and 6b show typical shapes of scattering resonances for the case of target nuclei of spin and spin •§• (indium). For the latter case, it must be kept in mind that the capture scattering involves a definite, single J value of the compound nucleus, while the potential scattering occurs for both possible J values of the system (in this case J = -§• ± ^ = 5 or 4), re- sulting in a finite scattering at the minimum. In addition, Fig. 6b cor- responds to a case where r « T y is> r„, and the presence of the radiative capture process also prevents the scattering minimum from going to zero. Another situation of interest is when r ~ T n i ^> T y , which is fre- quently encountered in nuclei of A < 50 for intermediate-energy neu- trons. In this case, application of Eq. (51) gives, for the position of the scattering minimum, r E r - E min = - ctn fc (53a) and, for the minimum cross section, (Tmm(n,n) = a-p (54a) 274 The Neutron [Pt. VII In addition, the maximum of the resonance is shifted toward higher energies: r #m a x - E r = - tan fc (55) and the peak cross section is <7 (n,n) = 4:irX 2 fi + <r p ' (55a) For a scattering resonance involving a target nucleus of 7 = and S scattering only (&> « £o) the scattering cross section will be zero at Fig. 6a. Schematic drawing of the cross section through a slow-neutron scattering resonance for I = neutrons, for a target nucleus of spin 0. A spin / ^ of the target nucleus would change this curve by multiplying it by /o(J) and adding to it a constant cross section [1 — fo(J)]4:irR 2 . the minimum (Fig. 6a). For 7 > or I > or both, the minimum will not fall to zero, since the potential scattering for angular momenta not associated with the resonance does not interfere with the capture scattering. Returning, now, to slow-neutron reactions in heavy nuclei (Eqs. 52), we note that the properties of a given resonance are completely deter- mined in terms of four parameters, E r , Y n , T y , and / . These can, in turn, be ascertained from four measured quantities: (1) the resonance energy, E r ; (2) the resonance width, T = T n + T 7 ; and (3), (4) the peak capture (or total) and scattering cross sections, <To(n,y) and <r (n,n). (We assume that R is given by r A Va X 10~ 13 cm.) Of these, c {n,n) is usually most difficult to measure, since (for r 7 y> r„) it is only a small fraction of the total. Thus, in a study of the 0.176-ev Cd 113 (7 = ^) resonance, Beeman (B12) has been able to extract all the parameters, and to show that the level of the compound nucleus has the spin J = 1. Sec. 2B] Interaction with Nuclei 275 In the absence of a measurement of o- ( n , n ), it is only possible to deter- mine the product / r„. For pure scattering resonances (r « r„ 55> T y ) the statistical factor / is immediately determined from the difference between the peak cross section and the minimum cross section, since c$ — tr m i n = 4irX 2 f. 20 - I - yur- J e = 4 - J c =5 -A ~ ' ' ^^s. ^= 4 -J Peak Values of (T sc are 764 x 10" 24 cm 2 (,/„=• 4) and 626xl0" 24 cra 2 (,/ c = 5). Resonance E r =1.44 ev. Jl J c =5 1 1 1 1 1 i .90 2.70 E (ev) Fig. 6b. The scattering cross section of indium (/ = %) for the 1.44 = ev reso- nance. The curves are computed from the known constants of the (capture) reso- nance and for the two possible values of J = 5 or 4. From Feshbach, Peaslee, and Weisskopf (F27). So far, we have discussed the case of a single resonance, neglecting the effects of all the other resonances. This approximation is valid in the vicinity of a single resonance when the level spacing D is large as com- pared to the level width r. If this is not the case, the levels can interfere with each other, provided they belong to the same class (same I, J, and parity). 1 Interference between levels has been discussed by a number of authors (B25, B24, P10, B61, S37, W29). Assuming, still, that there is only one resonance affecting the energy region in which we are interested, the cross sections far from the reso- 1 This statement applies only to the total cross section. Interference effects can always occur in the angular distribution of the reaction products provided, only, that the levels overlap. 276 The Neutron [Pt. VII nance, | E — E r | » r, are <r{n,a) = (56) (E - E r f a(n,n) ^ 4tR 2 (57) These are of especial interest in the thermal neutron energy region when the first resonance falls appreciably above or below thermal energies (I E r I ^> E fh ). In this case ,rX 2 /r K r g <rth(n,a) = — — — (58) Recalling that X ~ E~ Yi and, from Eq. (37), that r„ ~ E*, we have, provided that r„ is constant over the range of thermal energies, (59) where <r* is the cross section at a specific (arbitrary) energy E* (velocity v*). This is the well-known l/v law for slow-neutron reaction cross sec- tions far from a resonance. Deviations from the l/v law will result from too close proximity to a resonance; Eq. (56) shows that the effect of a close positive energy resonance (E r > 0) is that the cross section falls less rapidly than l/v, while a too-close negative resonance results in a thermal neutron reaction cross section which falls faster than l/v. The thermal neutron scattering cross section will, on the other hand, be constant if the first resonance is sufficiently far removed. However, in the case of scattering, the effects of interference between the potential scattering and the resonance will extend to considerable distances (Figs. 6a and 6b), so that even a far-away resonance can have an appre- ciable effect on the thermal neutron scattering. At the other extreme, we are frequently concerned with observations of cross sections which represent an average over many energy levels; this will be the case when the energy spread of the neutrons, used in the measurement, is large compared to the level spacing of the compound nucleus at the excitation energy involved. Assuming that X, r„, r are essentially constant over the energy spread of the neutron source, the average over Eq. (44) and sum over J = I ± % yield 2x 2 X 2 r„r„ a(n,a) = — (60) If states of I > are possible, the above average must be summed over all possible I values, each with its appropriate fi and appropriate neutron width T n i and level spacing D;. Sec. 2B] Interaction with Nuclei 277 If we consider heavy nuclei and 1 = interactions only, there are two regions of interest: (1) r « r r » r„; here, as for thermal neutron cap- ture, <r(n,y) oc 1/v; (2) r ~ V n » Y y ; in this range, <r{n,y) oc 1/E. The total cross section, however, has a more complicated energy dependence, since it is now necessary to take into account the potential scattering as well as the capture scattering and radiative capture (F28). The more general approach developed by Weisskopf and co-workers (F27, F28, W17, B34) gives, for the cross section for the formation of the compound nucleus, <r c « x(X + R) 2 P (61) The factor 7r(X + R) 2 is a measure of the area of interaction of the neu- tron and the target nucleus; P is the probability that the neutron will penetrate the nuclear surface. In the resonance region, P is a sharply varying function of the neutron energy, being the resonance factor. In terms of the simple picture which we have used, the incident neutron wave, of wavelength X = 1/k, must join smoothly on to the neutron wave, A = 1/K, at the surface of the nucleus. Since (for slow and intermediate neutrons) K^> k, the amplitude of the neutron wave in- side the nucleus will, in general, be very small, except for certain ex- ceptional values of the neutron energy at which the amplitude of the incoming wave at the nuclear surface has the maximum value. These special cases (energies) correspond to the resonances, for which the joining of the wave functions was schematically represented in Fig. 5. We have previously (Eq. 21) given an expression for P which can be interpreted as the average (over many resonances) probability for the neutron to penetrate the target nucleus. Hence, the average reaction cross section is _ 4tt(X + R) 2 kK c c ^ — - V- (62) (k + K) 2 For slow and intermediate neutron energies, X = 1/k » R, K = K >5> k, whence _ 4irX 2 & 4ir 500 , v <T C = = ~ — rr barns (62a) K kK E* (for E in electron volts). Thus, the average cross section for the forma- tion of the compound nucleus also follows a 1/v law in these energy ranges. Introducing the connection between neutron energy and neu- tron width from Eq. (37), irT n K k = (37a) 2D 278 The Neutron [Pt. VII the average cross section for the formation of the compound nucleus becomes - 2x 2 x 2 r n / ^ «*= ■ (63) Since the cross section a(n,a) = <r c T a /T, Eqs. (63) and (60) are seen to be equivalent, thereby confirming the interpretation of P (Eq. 21) as the average over many resonances of the penetrability. 4. Summary ; Energy Dependence of Neutron Cross Sections. In the preceding discussion we have outlined the basic considerations which determine the interactions of neutrons with nuclei. We have observed that the reactions of neutrons of a given energy with a given nucleus depend on the level structure of the compound nucleus, at the excita- tion energy resulting from the neutron capture. Although we are, at the present state of our knowledge, incapable of predicting the details of the nuclear level structure, a great deal can, nevertheless, be said concerning the average behavior of nuclei. In referring to neutron energies, we shall continue to use the classifi- cation of Section 1B7 : I. Slow neutrons: E < 1000 ev II. Intermediate neutrons: 1 kev < E < 0.5 Mev III. Fast neutrons: 0.5 Mev < E < 10 Mev IV. Very fast neutrons: 10 Mev < E < 50 Mev V. Ultrafast neutrons: 50 Mev < E In addition to the dependence of nuclear level properties on the kinetic energy of the incident neutron, the nuclear level spacing is very different for nuclei of widely different atomic weight, A. It is therefore convenient to classify nuclei according to the following system : I. Light nuclei : A < 25 II. Medium nuclei: 25 < A < 80 III. Heavy nuclei: 80 < A These dividing lines are, of course, not to be considered sharp bound- aries. While we shall return, in a subsequent section, to the experimental evidence on the question of the compound nucleus level spacing as a function of the nuclear atomic weight, it is useful to bear in mind the following summary of the average level spacing of the compound nucleus : (1) Light Nuclei. For slow and intermediate neutrons, the levels are very widely spaced, being ~1-0.1 Mev apart, on the average. Thus, Sec. 2BJ Interaction with Nuclei 279 very few levels are encountered, and these are relatively broad and easily resolved. In the fast and very fast neutron energy region, the levels are closer, but still separate and resolvable. (2) Medium Nuclei. For slow and intermediate neutrons, the levels are ~100-1 kev apart. Hence, few levels are encountered in the slow- neutron range, but many in the intermediate region. For fast neutrons the level spacing rapidly decreases, and the levels begin to overlap. For very fast neutrons the compound nucleus has a continuum of levels. (3) Heavy Nuclei. In the slow and intermediate neutron ranges, the level spacing is <~1000-5 ev. Many levels are observed for slow neu- trons; the levels are no longer resolved in the intermediate range; for fast and very fast neutrons the compound nucleus has a continuum of levels. For ultrafast neutrons and for all nuclei, the compound nucleus pic- ture requires considerable modification. (a) The Total Cross Section, c t . Most neutron cross section measure- ments involve the determination of the fraction of a given neutron beam transmitted through a known thickness of material; such measurements usually determine the total cross section, the sum of the cross sections for all processes (absorption, scattering) which remove neutrons from the beam, at = 2 <r(n,a) (64) a For a neutron flux of <j> (neutrons • cm -2 • sec -1 ) incident on a slab of material of density N (nuclei/cm 3 ) and thickness <(cm), the emerging, undeviated flux <$> is <j> = 4>o exp (-Nta t ) (65) Figure 7 shows typical slow-neutron total cross sections for the three categories of elements. The cross section of carbon exhibits thermal neutron interference effects due to the crystal structure of the graphite used in the measurement. Above ~1 ev, the cross section becomes and remains constant. The cross section of cobalt shows a scattering reso- nance at 120 ev. The cross section in the thermal neutron region has a 1/v component due to radiative capture. The indium and iridium cross sections show many resonances and large 1/v capture cross sections for thermal neutrons. Owing to the close spacing of the levels, the neutron width is small (Eq. 37) compared to the radiation width, so that radia- tive capture predominates in the observed resonances. For intermediate and fast neutrons individual levels have been well resolved only in light and medium nuclei; for these relatively large level spacings, the neutron widths are much larger than the gamma-ray 280 The Neutron [Pt. VII e 3 1 c <irai shite - 1 Sy mbol on Reference No. ^igure in Legend • 1 1 1 ® 3 1 1 l 4 l 1 I 4 I A 2 4 2 4 7 I 4 7 2 4 0.01 E„(av) 100 (1) Columbia Velocity Selector (unpublished). (2) W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev., 75, 1296 (1949). (3) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). (4) W. B. Jones, Jr., Phys. Rev., 74, 364 (1948). (a) 100 70 ^ 40 en | 20 13 10 7 T Co O r*J 2,500 t B.OOl) fc arns Symbol o Figure • o ® i Reference in Legerj 1 2 No. d 1 \ 1 3 / \ ■I \ \ / ""^. ->.... <A •iu^. y J \ / ' -.-.„ , r i \ / \ - — ' / 100 7 2 1000 4000 E n (ev) (1) Wu, Rainwater, and Havens, Phys. Rev., 71, 174 (1947). (2) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). (3) F. G. P. Seidl, Phys. Rev., 75, 1508 (1949). (b) Fig. 7. Typical total cross section vs. energy curves for slow neutrons, from Adair (A2). (a) Carbon, a light element. In this case, a t = <r sc - The variations, below ~0.04 ev, are interference effects due to the polycrystalline structure of the sample (see Section 5). (b) Cobalt, a medium element. Again, at ~ ct sc , except in the thermal neutron range. Effects of scattering resonances at 120 ev and >2000 ev are evident. Sec. 2B] Interaction with Nuclei 281 20,000 10,000 In 2 1000 Symbol on Reference No. Figure in Legend 1 2 3 ® 4 o 5 1 4 <A f i , ■1&T V. "-. ../ I 1 1 ^100 b" 7 4 2 10 7 4 2 1 ■". FT' - ■ I'l J £ ]aj I v. ■ — i u _ 4 7 2 0.1 2 4 7 2 1 10 B« (ev) 7 2 100 7 1000 (1) Borst, Ulrich, Osborne, and Hasbrouck, Phys. Rev., 70, 557 (1946). (2) Havens, Wu, Rainwater, and Meaker, Phys. Rev., 71, 165 (1947). (3) B. D. McDaniel, Phys. Rev., 70, 832 (1946). (4) E. Fermi and L. Marshall (unpublished). (5) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). (0) 4000 Ir j\ 1 1 1 1 1 1 2 11' J 'l Symbol on Reference No. Figure in Legend 1000 -tr • l x 2 7 ... / 1 " 2 S3 £ loo — ?** *-. l\ I -«** f f b ' — • P- '■^- ~~ S~ . V U- in i f 0.01 4 7 2 0.1 7 2 1 10 .E„(ev) 7 2 100 (1) Rainwater, Havens, Wu, and Dunning, Phys. Rev., 71, 65 (1947). (2) Sawyer, Wollan, Bernstein, and Peterson, Phys. Rev., 72, 109 (1947). (3) Powers, Goldsmith, Beyer, and Dunning, Phys. Rev., 53, 947 (1938). (d) Fig. 7 (Continued), (c) Indium, a heavy element. The famous resonance, at 1.44 ev, is well resolved. There are probably many unresolved resonances above 10 ev. (d) Iridium, a heavy element. The first two capture resonances are fairly well re- solved. There are probably many unresolved resonances above 20 ev. 282 The Neutron [Pt. VII widths, and the observed resonances are due to scattering. We post- pone the discussion of scattering resonances and consider, instead, the average behavior of the total cross section for intermediate and fast neutrons. The average total cross section, according to the theory of Feshbach and Weisskopf (Eqs. 49, 61, 62), depends on the properties of the nucleus involved only through the nuclear radius, R. 1 ft! 2.0 1.0 -1 1 " Uo = B -.1 4 - y _\ - As V s 2 *" \ cje+i ) 2 - <^ ii; ^ _ — - 123456789 10 x — kR Fig. 8. Total cross sections (averaged over resonances) as a function of x = kR, for Xo = KqR = 5 and 8. <ro is the total cross section for an impenetrable sphere of radius R. The broken curve gives the approximate behaviors for large x. From Feshbach and Weisskopf (F28). Figure 8 is a plot of the expected average total cross section vs. neutron energy. The cross section is given in units of wR 2 , and the energy in units of x = kR = 0.222RE' A (66) where R is in units of 10~ 13 cm and E in Mev. The curves are plotted for two values of X = K R: X = 5 (A « 40) and X = 8 .(A ~ 160). The curve labeled <r is the total cross section of an infinitely repulsive 1 The parameter K «1X 10 ~ 13 cm, the wave number of a neutron in the nucleus, is assumed to be the same for all nuclei; this is not strictly true, but the variations in Ka are expected to be small. Sec. 2B] Interaction with Nuclei 283 sphere of radius R. For large values of x, the curves are approximately- fitted by the broken curve, «r« « 2t(R + X) 2 (67) It is important to note that, while the value of the total cross section approaches 2icR 2 for X « R, the deviations from this asymptotic value are quite appreciable, even for very fast neutrons; thus, for E = 50 Mev, X = 0.64 X 10 -13 cm, which cannot be neglected. In Fig. 9 the experimental values of the total cross section are com- pared with the theory for a number of elements: (a) iron, A « 56; (b) silver, A ~ 108; (c) antimony, A « 122; and (d) lead, A « 207. The measurements employed relatively poor resolution, and as a result the averaging process was, to a large extent, performed experimentally. The agreement with theory is seen to be excellent, except in the case of antimony, where the observed cross section falls considerably below the theoretical value at low energies. 1 (b) Cross Section for Formation of a Compound Nucleus, a c . The cross section for the formation of a compound nucleus includes all reactions, except elastic scattering in which the quantum state of the nucleus is unchanged. It is usually defined as <?c = O t — (T e l (68) While this definition does not accurately take into account the effects of interference between capture and potential scattering, such effects are essentially eliminated when the cross section is averaged over many resonances. For light and medium nuclei, a c for slow and intermediate neutrons is mainly due to capture scattering (except for thermal and slower neu- trons). For heavy nuclei, in these energy regions, a c is almost entirely due to radiative capture. For fast and very fast neutrons, elastic scat- tering contributes little to the value of <r c , the other possible reactions — inelastic scattering, (n,p), (n,a), (n,2n), etc. — predominating. The energy dependence of a c (averaged over resonances) is described by Eqs. (61), (62), (62a) and plotted in Fig. 10. (The method of plot- 1 This "anomalous" type of behavior has been found, by the Wisconsin group (M31), to occur with unanticipated frequency. Furthermore, extension of the total cross section measurements to 3.2 Mev has brought to light, for the heavier ele- ments, the frequent existence of a broad maximum in a t which appears to move to higher energies with increasing mass number. These phenomena are strongly sug- gestive of the scattering by a finite potential, and could possibly be interpreted as supporting an "independent particle" model for fast neutron scattering. It is, at present, difficult to see how these results can be reconciled with the apparent wide- spread applicability of statistical models. 284 The Neutron [Pt. VII Fe 10 n _ OS N 6 " k /i 4 "I mi "i ^ "~-fV Theory V ), ?, Experin ent i i t i 1 1 1 1 1 0-2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.1 x=kR (a) t3 Ag - -~"T - _ ^- eory - Exp J iriment P^ rV '° sr \7 i 1 I i 1 1 1 1 1 i 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 x=kR (b) Q5 Sb ~^L 0.2 0.4 0.6 0.8 1.0 x=kR 1.2 1.4 1.6 1.8 2.0 (C) Fig. 9. Comparison of the theory of Feshbach and Weisskopf (F28) with measured total cross sections for a number of elements. The measurements are due to Bar- schall and co-workers (references in F28). (a) Iron, A =< 56. (b) Silver, A « 108. (c) Antimony, A «= 122. In the case of antimony the observed cross section is in better agreement with the prediction of scattering from a rigid sphere. Sec. 2B] Interaction with Nuclei 285 8 Pb 6 ~~~- — ~v_ . Theory <* - Experiments^ 2 i i ' | 1 1 1 1 1 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 x=kR (d) Fig. 9 (Continued), (d) Lead, A = 207. Sd.b 2.4 2.2 2.0 1.8 1.6 1.4 1.2 I Vo = 5 \ • -*) 2 V ^0=8 ^ "^ - :;:: ^=c: ^^5--. - — . 1.0 ^cii 10 Fig. 10. Cross section for the formation of a compound nucleus vs. x = kB, accord- ing to Feshbach and Weisskopf (F28). The solid curves are for X = KoR = 5, 8, and 11. The broken curve, appropriate to large x, is <r c « -^a t « ir(iJ + X) 2 . 286 The Neutron [Pt. VII ting is the same as that employed in Figs. 8 and 9.) As seen from Eq. (62), a c -> ir(R + X) 2 for x » X ; the deviation of <r c from w(R + X) 2 depends on the value of x/X , being «11 percent for x = X and «4 percent for x = 2X . For large values of x, <y c ~ \<j t and, since cap- ture elastic scattering is negligible for large x, a c ~ a e i; i.e., for large x, the total cross section divides into approximately equal parts elastic scattering and capture. C. Types of Neutron Reactions In the previous section, we have discussed the general properties of neutron reactions and the formation of the compound nucleus. Although it has been observed that the nature of the compound nucleus de-excita- tion is governed by the competition between the various possible modes of decay, the properties of this competition have been treated in the most general way. This section is devoted to the anatomy of neutron reactions — to the details of the competition between the various possible neutron reactions, as they are influenced by the properties of the nuclei involved and by the energy of the impinging neutrons. The relative probability of a given neutron reaction (n,a) — neutron in, particle a out — is defined in terms of the cross section <r(n,a). For a (pure) material containing NV nuclei in a neutron flux <£, the number of (n,a) reactions per second is NV<f>a-(n,a) . 1. Scattering (n,ri). Neutron scattering is one of the two reactions which are energetically possible for all nuclei at all neutron energies. (The other is radiative capture.) The observed elastic scattering is, in general, the result of a superposition of the potential scattering and the capture scattering (Eq. 51). However, this superposition is such that it is coherent (addition of amplitudes, interference effects) for the capture scattering and that part of the potential scattering which involves the same neutron orbital angular momentum, Zh, and the same total angular momentum state, /. The remainder of the potential scat- tering is observed as an incoherent background (addition of cross sec- tions) to the scattering, as described by Eq. (51). (a) Slow Neutrons. In the slow-neutron energy range, the neutron wavelength is very much larger than any nuclear radius. Hence, neu- trons can react (with appreciable probability) only in the I = state. Thus, the potential scattering cross section is 4irR 2 (Eq. 49a), and the scattering cross section reduces to Eq. (52). In the absence of nuclear resonances, the slow-neutron (potential) scattering cross section would be expected to increase with atomic num- ber as A % . Table 5 is a compilation of observed slow-neutron scatter- ing cross sections together with the corresponding values of 47r.fi! 2 , com- Sec. 2C] Interaction with Nuclei 287 puted on the assumption R = 1A7A H X 10 -13 cm. The scattering cross sections, in many cases, deviate considerably from 4irR 2 , being both larger and smaller. These deviations are too large to be ascribed to fluctuations in R. An additional, noteworthy fact is that the deviations occur, with comparable frequency, at all values of the atomic number. These fluctuations in <r sc can be ascribed to the effect of a nearby resonance, being the result of the interference between the potential and the resonance scattering. A value of <r sc > 4xi2 2 indicates that the closest resonance is at an energy lower than that at which the cross sec- tion was measured (frequently E r < 0), while <r sc < 4irR 2 indicates a resonance at a higher energy. The probability that a scattering measurement, made at a given (arbitrary) neutron energy, will fall on a resonance is inversely propor- tional to the level spacing. That is, the average distance of a given energy from a resonance is proportional to the level spacing; thus, the larger the level spacing, the farther, on the average, from the closest level. However, the region over which interference effects are impor- tant is also proportional to the level spacing (Eq. 53). Hence, the probability that a scattering measurement will give a cross section dif- ferent from A-wR 2 is essentially independent of the level spacing and, therefore, of the atomic number of the scatterer. TABLE 5 Slow Neutron Scattering Cross Sections for Free Nuclei Element At Csc t (barns) 4irfl 2 § (barns) E || (ev) References If iH 1 20.36 W10, M24 iH 2 3.3 th S33 2 He 4 1.4 (0.2) th H41 3 Li 7 ~1.5 0.99 th S33 4 Be 9 6.1 1.17 (0.8) th S33 B B 11, 10 3.9 1.33 (1.5) C W10 6 C 12 4.70 1.42 (1.8) th W10 7 N U 9.96 1.58 10-200 W10, M25 8 16 3.73 1.70 (2.3) 15-1000 W10, M25 9 F 19 3.3 1.93 0.25-40 W10, R5 288 The Neutron [Pt. VII TABLE 5 (Continued) Slow Neuteon Scattering Ceoss Sections toe Free Nuclei Element At <Tsc t (barns) 4irj? 2 § (barns) E || (ev) References If ioNe 20 2.4 2.01 th H41 u Na 23 3.3 2.20 1-800 H58 i 2 Mg 24, 26, 25 3.9 2.28 (2.5) th S33 «A1» 1.35 2.44 (2.7) th W10 uSi 28 2.2 2.51 1-100 W10, R5 i 6 P 31 ~3.3 2.68 1-10 G20 16S 32 ~1.2 2.74 (2.1) 10-400 A2 nCl 35,37 14.2 2.93 (2.8) th H59 isA 40 0.8 3.17 th H41 i 9 K 39 ~2 3.13 th S33 2oCa 40 3.3 3.18 th S33 2lSc« 12.8 3.44 th H40 2 2 Ti 48 6 3.58 th S33, F27, G12 23V 61 5.02 3.73 th H26 24Cr 52 3.7 3.78 th S33 24Cr 63 8.4 3.83 th H40 26 Mn 56 2.1 3.93 th S33 26 Fe 54 2.4 3.88 th S33 2 6 Fe 56 12.5 3.97 th S33 26 Fe 67 2 4.02 th S33 2eFe 56 11.3 3.97 th S33 11.1 (3.9) 1.44 F27, H36 27C0 69 5 4.12 C F27, B57 28 Ni 68 27 4.07 th S33 28Ni 6 ° 1.0 4.16 th S33 28 Ni 62 9 4.25 th S33 28 Ni 58, 60 16.7 4.10 th S33 29CU 63, 65 7.6 4.33 th S33 8.3 (3.8) 1.44 F27, H36 3oZn 64, 66, 68 4.1 4.41 (4.4) th, 1.44 S33, F27, H36 aoZn 67 7 4.48 th H40 3iGa 69, 71 3 4.60 th W10 32Ge 74, 72, 70 8.3 4.71 th S33 33AS 76 ~7 4.83 th S33 Sec. 2C] Interaction with Nuclei 289 TABLE 5 (Continued) Slow Neutron Scattering Cross Sections for Free Nuclei Element A t 08C X (barns) 4ttK 2 § (barns) E || (ev) References f 34 Se 80, 78 10 5.00 (5.0) C F27, G12 ssBr 79, 81 5.9 5.04 th S33 3eKr 84, 86, 82, 83 7.2 5.20 th H41 37 Rb 85,87 5.4 5.27 th S33 3sSr 88 9.3 5.36 th S33 39Y 89 3.9 5.41 th H40 40 Zr 90, 94, 92, 91 ~7 5.50 th S33 4lNb 93 6.1 5.58 th S33 42 Mo 98, 96, 95, 92 7.3 5.69 th S33 44R11 102, 104, 101, 99, 100 6 5.92 C F27, G12 4 5 Rh 103 3.5 5.96 th H40 46 Pd 106, 108, 105, 4.7 6.11 th S33 110 4.4 C F27, G12 4 7 Ag 107 10 6.12 th S33 47Ag 109 6 6.20 th S33 4?Ag 107, 109 7 6.15 (5.8) th S33 4 8 Cd 114, 112, 111, 110, 113 5.3 6.32 (6.5) 5-100 G20 49ln 115 2.2 6.42 th W10 soSn 120, 118, 116 4.8 6.56 (6.9) th S33 6 iSb 121, 123 4.1 6.67 (6.7) th, 0.1-2 S33, G20 5 2 Te 130, 128, 126 5 6.88 C G12 53I 127 3.7 6.86 th S33 5 4 Xe 132, 129, 131, 134 4.3 7.02 th H41 65 Ce 133 ~7 7.08 th S33 56Ba 138, 137 8 7.23 C G12 69 Pr 141 7.9 7.35 th H40 62Sm 152, 154, 147, 149, 148 23 7.68 th H40 63EU 153, 151 29.7 7.73 th H40 64Gd 158, 160, 156, 157, 155 26 7.90 th H40 290 The Neutron [Pt. VII TABLE 5 (Continued) Slow Neutron Scatteeing Cross Sections for Free Nuclei Element A\ f«c t (barns) 4xfl 2 § (barns) E || (ev) References H 72H1 180, 178, 177, 179 25.7 8.61 th H40 73 Ta 181 6.9 8.69 th S33 74W 184, 186, 182, 183 5.6 8.78 th S33 74 W 186 23 8.84 th H40 76 0s 192, 190, 189, 188 ~10 8.98 C G20, G12 7sPt 195, 194, 196 11.1 9.14 th S33 79 AU 197 ~9 9.19 (7.1) th S33 soHg 202, 200, 199, 15 9.30 0.1-10 G20 201, 198 21.5 (8.7) th H60 81T1 205, 203 9.6 9.42 0.1-1 F27, G20 82 Pb 208, 206, 207 11.5 9.51 th S33 12.4 (7.6) 1-10 F27, G20 83 Bi 209 9.2 9.56 5-10 F27, G20 10 (7.8) th S33 92U 238 8.2 10.5 th U2 f When more than one isotope is involved, this column lists the most impor- tant isotopes, of relative abundance greater than 10 percent, in the order of abundance. t The cross sections are for free nuclei. Most observations have been made on bound nuclei; these have been corrected according to the relationship °"free -C-*i)' ^bound § Measured nuclear radii, taken from the collection of Blatt and Weisskopf (B34), are given in parentheses. Otherwise they are computed on the basis of R = 1.474H X 10- 13 cm || The symbols have the following meanings: th, a thermal Maxwell distribu- tion of neutrons at a temperature of <~300°K; C, those neutrons, in a thermal distribution, capable of penetrating through appreciable thicknesses of cadmium. (These have energies above the cadmium cut-off, ~0.5 ev.) f This compilation leans heavily on the previous compilations of Shull and Wollan (S33), Way at al. (W10), and Feshbach, Peaslee, and Weisskopf (F27), in which references are given to the original investigations. Sec. 2C] Interaction with Nuclei 291 However, the curves of c sc vs. E will be very different for nuclei of widely different atomic number, since the number of resonance and their characteristics depend strongly on the level spacing. (1) Light Nuclei. In general, the main reaction of slow neutrons with light nuclei is elastic scattering. There are four exceptions — Li 6 and B 10 , for which the main reactions are (n,a); He 3 and N 14 , for which there is an appreciable slow-neutron (n,p) cross section. Aside from these, a sc = v t (see Fig. 7a). (2) Medium Nuclei. With regard to the scattering of slow neutrons, medium nuclei are similar to light nuclei, except that resonances are sometimes encountered. These are primarily scattering resonances since the level spacing is quite large. For this case r ~ r n > r T , the peak cross section 2.6/ <r (n,n) ^ 47rX r 2 / = X 10 6 barns (55a') E r (E r is in electron volts.) The 120-ev resonance in cobalt (Fig. 7b) is an example of such a scattering resonance. The peak cross section for this resonance should be ~13,000 barns since, for Co 59 , I = f-(/~ f). The smallness of the peak cross section in Fig. 7b, as well as the absence of the expected interference between the resonance and potential scat- tering, is due to the poor resolution of the measurement. Since the TABLE 6 Properties of Some Neutron Scattering Resonances Com- Target Nucleus Spin / E r (ev) r„ (ev) r„/r pound Nucleus Spin J Observed <r (barns) 4xX r 2 / (barns) References nNa 23 3 2 3,300 340 ~0.999 2 550 540 H58, t 16 S 32 111,000 18,000 ~l 1 2 21.8 23.4 Al 17C1 36 2 -75 2.63 fl 0.90 t (2) H59 2 6 Mn 55 345 20 0.990 3 4-5,000 4,400 S17, H40, H43 27C0 59 7 2 123 »4 0.94 (4) 12,500 12,200 H38, S17, H40, t 33AS 76 3 46 0.11 0.72 ~28,000 t 69ft 141 5 1 (-10?) 0.93 (~130,000) H40 6 2 Sm 152 8.2 ~0.3 0.66 1 2 320,000 H40 74W 186 19.25 0.25 0.62 1 2 ~90,000 135,000 H39, H40, S19 8lT1 eon) 1 260 «3.2 0.52 (1) 7,500 H40 83Bi 2M "Z 770 3.5 ~1 ~1,700 t t Some of the original data in this table have been revised in the light of recent measurements by the Harwell time- of-flight velocity selector group (M26) and by the Argonne fast chopper group (whose members included L. M. Bol- linger, R. R. Palmer, and S. P. Harris). We are grateful to these groups for private communications of their results. t These values are at E = \ E r \ =75 ev. 292 The Neutron [Pt. VII recognition, by Goldhaber and Yalow (G14), that the resonance of manganese at ~300 ev is a scattering resonance, many such resonances have been observed and investigated in medium (and some heavy) nuclei. Table 6 summarizes the properties of a number of scattering resonances. (3) Heavy Nuclei. In heavy nuclei, with their relatively small level spacings, the gamma-ray width is generally larger than the neutron Symbol o Figure 1 1 l Reference No. in Leo-end 1 ® 1 oA 2 : - 4 1 k~ L «h < o i ^ ® / 1 // •V i ® V 9 . l*C / ***•& w A A A ' 0.2 0.4 0.G 0.8 1.0 1.2 1.4 1.6 1.8 £„(Mev) (1) Fields, Russell, Sachs, and Wattenberg, Phys. Rev., 71, 508 (1947). (2) Adair, Barschall, Bockel- man, and Sala, Phys. Rev., 75, 1124 (1949). (3) Bockelman, Miller, Adair, and Barschall, Phys. Rev., 84, 69 (1951). (4) Freier, Fulk, Lampi, and Williams, Phys. Rev., 78, 508 (1950). Fig. 11. Total (scattering) cross section of O 16 for intermediate and fast neutrons, showing well-resolved P-wave resonances at 0.440, 1, and 1.3 Mev; from Adair (A2). Measurements from ~1.8 to 4 Mev have exhibited additional resonances whose angular momenta and parities have been deduced by Baldinger, Huber, and Proctor (B5) by means of accurate measurements of the angular distributions of the elastically scattered neutrons. width, r ~ T 7 > r„. Accordingly, the peak scattering cross section is small compared to the peak radiative (and total) cross section, <ro(n,n) = ~ <r (n,y) (47a) The resonance scattering of cadmium (B12) is an example of this case. Owing to fluctuations in the nuclear level spacing, and to the differ- ences in neutron binding energies, levels are sometimes encountered in heavy nuclei for which T n > F y . The broad resonance of W 186 at 19 ev has been found to be such a scattering resonance (H39). 1 1 The existence of broad, scattering resonances for slow neutrons on heavy nuclei might be additional evidence for the necessity, under certain circumstances, of re- garding neutron reactions from the independent particle point of view. Sec. 2C] Interaction with Nuclei 293 (su-req) % D u o <y &H d E- X E-i CD d '55 (H T3 fe d d 03 a> OJ •d o d d +-> oi (=1 o d d o tn CD 05 ffl ri ^ t-. T3 to =3 > .id O Eh d F-H ol 0) CO d rd N ,d BQ (1 bl) c« d O d o £ a o ro o. o d o Eh 03 r/> 0> fl 0) 1-1 CQ +s h d > -d 3 o n d a> 0J a; 1— I o J4 o 03 t3 *Q H Eh 3 H 0J +3 E- o to < ri u < u 03 3 0) a> r^ M o O a CO V) .2 o £ op O o o go d 03 en CO I7J v ;__; a O 03 03 a M) V) o d 0> « -d -t-> O « tf +■* si CO d "3 r— i T3 03 n O -M H CJ ri d a, c3 a O a; is hr -m Pn Q .fi 294 The Neutron [Pt. VII (siueq) ?x> Sec. 2C] Interaction with Nuclei 295 (b) Intermediate Neutrons. The intermediate neutron energy region is, with respect to the interaction of neutrons with nuclei, a transition region. At the low-energy end, the neutron wavelength is still large V i-1 ' V \ » v VVi \\ A 1 l\ y v \-> s f\A' y )n ■\h[ Ws o°A o o A A 0.4 0.6 £7„(Mev) 1.0 Fig. 13b. Scattering cross section for intermediate-energy neutrons of V 51 ; data from Blair and Wallace, Phys. Rev. 79, 28 (1950). S 3 s n n Symbol o Figure n Reference N in Legend 1 •1 1 If J L • 3 / / / i '1 \ \i '■Ahh { Ute i / v\ « \i \ i A I A 0.7 0.E 1.2 1.4 E„(Mev) 1.6 1.8 (1) Peterson, Barschall, and Bockelman, Phys. Rev., 79, 593 (1950). (2) Freier, Fulk, Lampi, and Williams, Phys. Rev., 78, 508 (1950). (3) Fields, Russell, Sachs, and Wattenberg, Phys. Rev., 71, 508 (1947). Fig. 13c. Scattering cross section of S 32 in the fast-neutron region; from Adair (A2). compared to nuclear radii ; at the upper end, the neutron wavelength is comparable with (and, for heavy nuclei, smaller than) nuclear dimen- sions. In this region, the character of the neutron scattering process slowly changes. (1) Light Nuclei. Figure 11 shows the scattering cross section of oxygen (7 = 0) in the intermediate (and lower end of the fast) energy range. The cross section is constant up to the resonance at 440 kev. 296 The Neutron [Pt. VII This resonance must be ascribed to capture of I = 1 neutrons (P reso- nance), since its peak cross section (14 barns) is greater than 47rX r 2 = 6 barns, and since it shows none of the interference properties associated with <S resonances. The value of the peak cross section is consistent with a value J = -J (/i = 2) for the compound nucleus; i.e., a a (n,n) = 87rX r 2 + <r p ' = 12 + 4 = 16 barns. The resolution of the measure- ment is excellent (~10 kev), but its finite value is sufficient to account for the difference of ~2 barns between the observed and expected peak cross sections. (2) Medium Nuclei. In Fig. 12 is plotted the total (scattering) cross section for sulfur {A = 32). This curve is especially interesting, since it illustrates many of the features of scattering resonances which we have previously discussed. The broken curve is a plot of 4xX 2 . The three *S-wave resonances, at 111 kev, 375 kev, and 700 kev, have peak cross sections close to 4ttX 2 , indicating the excellent resolution of the measurements. They show very well the interference between the po- tential and resonance scattering. The other six resonances appear to be due to P(l — 1) or D(l = 2) neutrons, and are incompletely resolved. Of these, the 585-kev resonance can be ascribed to a / = f- state of the compound nucleus, S 33 (P13). (3) Heavy Nuclei. For heavy nuclei in the intermediate-energy re- gion, the resonances are usually much more closely spaced than the resolution of the best available measuring devices. At the low-energy end the resonances are still primarily due to radiative capture (T y > T n ), while at the upper end the neutron width has caught up with the gamma- ray width, and scattering predominates. Toward the high-energy end of the intermediate region inelastic scattering starts to compete as a possible reaction, but we postpone the discussion of this reaction. For certain of the heavy nuclei, the level spacing of the compound nucleus, at the excitation energy corresponding to neutron capture, is anomalously large. Such appears to be the case for the magic number or near magic elements lead and bismuth, owing either to an anomalously small neutron binding energy, or to an intrinsically large level spacing, or both. The cross section of Pb 206 is shown in Fig. 13a. As an interesting comparison, we also plot the cross section of V 51 in the intermediate region (Fig. 13b), and that of S 32 in the fast-neutron region (Fig. 13c). All of these show approximately the same level spacings. The curves of Fig. 13 can be used to illustrate some other interesting possibilities in scattering resonances. If the resonances are still resolva- ble at relatively large values of x = kR >" 1 (due either, as in S 32 , to large spacing for fast neutrons or, as in Pb 206 , to a large nuclear radius), I = 1 potential scattering becomes appreciable, and P-wave resonances Sec. 2C] Interaction with Nuclei 297 are no longer expected to be symmetrical. Thus, the 1055-kev sulfur resonance must, because of the large peak cross section, be ascribed to neutrons of I > 0; however, it also shows a definite interference mini- mum, due to the P-wave potential scattering. It is therefore a Py 2 reso- nance. Another interesting anomaly occurs in the region £ = % ~ v/2. (&>o ~ f/2, for which the same considerations apply, requires neutrons of much greater energy.) For an <S-wave resonance in this energy region, application of Eqs. (53a) and (55) indicates that the minimum cross section, o- min = a p ' , occurs at E min = E r , while the maximum disappears (i.e., -E/max — »• °°). This leads to cross-section curves which have the appearance of inverted resonances, as observed in the Pb 206 cross sec- tion of Fig. 13a. (c) Fast Neutrons. Of the total cross section for fast neutrons, ap- proximately half involves the formation of a compound nucleus; the rest is due to elastic (potential) scattering, in which the neutron passes close to, but not into, the target nucleus. I. Capture Scattering and Inelastic Scattering With the exception of a few light nuclei (and the heaviest nuclei) the most probable result of fast-neutron capture is neutron re-emission. As long as the incident neutron energy is less than the energy of the first excited state of the target nucleus, the neutrons will be re-emitted with their incident energy (minus, of course, the energy lost to the recoil nucleus to conserve momentum). As soon as the incident neutron energy exceeds the energy required for the excitation of the lowest level of the target nucleus, it becomes possible for the product nucleus to be left in the excited state; correspondingly, the emitted neutron will have a smaller kinetic energy than the incident neutron. This process is known as inelastic scattering. From the point of view of the excited compound nucleus, decay to the ground state (capture elastic scattering) and to the various energetically possible excited states (inelastic scattering) of the product nucleus are competing processes. Each mode of decay is characterized by a partial width, r,-, which is a measure of its relative probability. However, from the experimental viewpoint, there are, in addition to the difference in energy of the elastically and inelastically scattered neutrons, two dis- tinguishing features between the two processes: (1) Capture elastic scattering and potential scattering are coherent processes, and the elas- tic scattering cross section can therefore exhibit interference effects of the type previously discussed. (2) Inelastic scattering leaves the prod- 298 The Neutron [Pt. VII uct nucleus in an excited state, from which it decays by the emission of one or more gamma-rays. Indeed, it was through the observation of this gamma-radiation that the inelastic scattering process was dis- covered (Lll, K12). There have been a number of subsequent investi- gations of inelastic scattering by the observation of the resulting gamma- radiation (A29, S12, B13, G27); however, all but the most recent of these have been too crude to permit detailed interpretation. The most useful information concerning inelastic scattering has been derived from experiments involving the observation of the energy dis- tribution of inelastically scattered neutrons. The interpretation of such experiments has been discussed by Feld (F9). In the ideal inelastic scattering measurement, a monoenergetic neutron beam would be scattered from a thin target of the element under in- vestigation; the scattered neutrons would have a line spectrum, each line corresponding to the excitation of a given level of the target nucleus. The energies of the inelastically scattered neutron groups would give the positions of these energy levels, and the relative strengths of the groups would be a measure of the relative values of the partial widths for the corresponding compound nucleus decay. In addition, the angu- lar distribution of a given group of inelastically scattered neutrons (with respect to the direction of the incident neutrons) would yield valuable information concerning the angular momentum and parity properties of the levels involved. Finally, each group of inelastically scattered neutrons is accompanied (in coincidence) by one or more gamma-rays; the correlation between the direction of emission of the neutrons and gamma-rays would yield further information concerning the angular momentum and parity properties of the levels. An experiment of the type outlined above is, in practice, exceedingly difficult, since it requires, in addition to a monoenergetic neutron source, a fast-neutron detector of high resolution. Most attempts to measure the energy distribution of inelastically scattered neutrons have em- ployed "threshold" detectors, i.e., a detector sensitive only to neutrons of energy above a fixed value, E t . Since, in most cases, the sensitivity function of the detector is not well known, such experiments are usually difficult to interpret unambiguously. Many inelastic scattering measurements have employed heteroener- getic neutron sources, such as the Ra-a-Be or Ra-a-B source, and thresh- old detectors. Owing to the extreme difficulty in interpreting such ex- periments (S65), they will be omitted from this discussion. (1) Light Nuclei. The low-lying levels of light nuclei are very widely spaced (~0.5-5 Mev). Hence, fast neutrons can excite few levels. In many cases, for neutrons of a few Mev, only one level of the target Sec. 2C] Interaction with Nuclei 299 nucleus will be available for excitation. The inelastic scattering cross section will depend in detail on the properties of this level, according to the following general considerations: Let <r c be the cross section for the formation of a compound nucleus. 1 The cross section for the excitation of the level (through inelastic scattering) is <r z = — (42a) Ti, the partial width for inelastic neutron emission, is given by Eqs. (37) to (40) : T x « (21 + ^CDiE^vi^R) (38a) in which the symbols have their usual meanings; E\ and ki are the energy and wave number of the (inelastically) scattered neutron : E^E - D x (33a) (E is the incident neutron energy; D x is the excitation energy of the target nucleus level involved.) The total width r = r + t 1 + r a + r 6 + • • • (28a) includes the width for elastic neutron re-emission, r , and the widths for decay by emission of any other particles, r , Tb, etc. In the expression for a\ (Eq. 42a) all the factors are, to a greater or lesser extent, energy dependent. However, in the immediate neighbor- hood of the threshold (E « Dx), the main energy dependence arises through the factor Ei A vi{kiR) in IV (It is clear that, for energetic reasons, Ti = for E < D x .) If we consider a region close to the threshold, such that E x <<C E, the factors a c and r are essentially con- stant. For inelastic scattering in which S-wave (I = 0) neutrons are emitted, the cross section will have an E^ A energy dependence near the threshold; for emission of I > neutrons, the cross section near thresh- old will increase as E^ l+y2) . The angular momentum of the emitted neutrons is determined by the values of the spins and parities of the initial (compound nucleus) and final (product nucleus) states. In many cases these values are such that neutrons of only a single, definite angular momentum can be emitted. In some cases, however, the spin values are such that angular momen- tum conservation could be satisfied in a number of alternative modes of decay, i.e., more than one I value is possible for the emitted neutron. 1 The energy dependence of a c has been discussed on page 283. It should, however, be borne in mind that the general considerations, used to derive Fig. 10, are of ques- tionable validity for light nuclei in the fast-neutron range. 300 The Neutron [Pt. VII In such cases the parity selection rule further limits the number of pos- sible I values; e.g., if the parity of the initial state is the same as that of the final state, neutrons can be emitted only with even I values; for states of opposite parity, only odd I values are permitted. Consequently, and also because of the strong energy dependence of Ti, one — the lowest — of the possible I values of emitted neutrons will be predominant in most cases. When the angular momentum of the emitted neutrons has a definite value and if, furthermore, the absorption of the incident neutrons to form the compound nucleus state involves a single I value, the inelasti- cally scattered neutrons will have a definite and predictable angular distribution with respect to the direction of the incident neutrons. The angular distributions, in a number of possible cases, have been given by Hauser and Feshbach (H44) . As an example of the type of angular distribution that could be en- countered, we give the result for a specific (hypothetical) case: Target nucleus: I = \ Compound nucleus : J = 3 Product nucleus: /'=■§• Incident neutron: I = 2 Scattered neutron: V = 1 The resulting angular distribution of (inelastically) scattered neutrons is W(e) = i + |f cos 2 e (0 is the angle between the incident and scattered neutrons.) Similar information concerning the three states involved could be obtained by observation of the angular correlation between the inelasti- cally scattered neutrons and the gamma-radiation which follows the inelastic scattering. In the example given above, the transition AI = 1 with parity change would be achieved through the emission of an electric dipole (or magnetic quadrupole) gamma-ray. Angular correlations in successive particle gamma-ray emission are discussed in Part IX of Volume III. As previously indicated, many cases of inelastic scattering will be considerably more complicated than the example discussed above. For instance, it may be possible for the incident neutrons to excite more than one compound nucleus state, requiring a number of incident neutron I values. Correspondingly, the decay of the compound nucleus to the excited product nucleus can also involve a number of different Sec. 2C] Interaction with Nuclei 301 angular momenta, and the angular distributions for the different I values can interfere. If this situation prevails, the calculation of the angular distribution of the inelastically scattered neutrons requires further knowledge of the details of the competition. Some of these possibilities have been considered by Hauser and Feshbach (H44) and by Wolfenstein (W38) : There has been very little experimental investigation of the inelastic scattering of fast neutrons by light nuclei. The work of Beghian, Grace, Preston, and Halban (B13, G27), on the inelastic scattering of 2.5-Mev neutrons by beryllium, carbon, fluorine, magnesium, sulfur, chromium, iron, and copper is of interest in connection with the above considera- tions. They detected the inelastic scattering by observing the resulting gamma-radiation and used the intensity of the gamma-rays as a meas- ure of the cross section. In the cases of carbon and beryllium they were unable to detect any gamma-radiation. For the other elements they observed a single (monoenergetic) gamma-ray in the first four cases, indicating that only a single level of the target nucleus was in- volved; the last two yielded complex gamma-ray spectra. These results are summarized in Table 7a. The observed cross sections, when combined with a knowledge of <r c , yield values of the relative probability for inelastic scattering Ti/T (Eq. 42a). These are shown for fluorine, magnesium, and sulfur in Table 7b, together with the data from which they were derived. The values of cr c have been estimated from the available data (A2, W32). In the last column of the table, we also give theoretical values of Ti/T, based on the assumption that only elastic and inelastic scattering are of importance (r = r + 1^), and that both types of scattering in- volve only I = neutrons. In the case of magnesium the inelastic scattering has been observed directly by Little, Long, and Mandeville (L20). They scattered 2.5-Mev neutrons (from a D-D source) in a block of magnesium and measured the energy distribution of the scattered neutrons by observing proton recoils in a cloud chamber. In addition to the elastically scattered group, they observed a single group of inelastically scattered neutrons whose energy corresponds to an excited state of magnesium at 1.30 Mev (compare Table 7a). From the strengths of the two groups, they com- puted values of ~1.6 barns and ~0.6 barn for the elastic and inelastic scattering cross sections, respectively, in excellent agreement with the result of Grace et at. and the measured value of c t ~ 2.2 barns. The value of Ti/T, shown in parentheses in Table 7b, is computed from their value of <ti « 0.6 barn. 302 The Neutron [Pt. VII TABLE 7 (a) Results of Grace, Beghian, Preston, and Halban (B13, G27) on Inelastic Scattering of 2.5-Mev Neutrons, from Observation of the Resulting Gamma-Radiation Element Atomic Weight Observed E y (Mev) Observed a\ (barns) Be 9 <0.014 C 12 <0.006 F 19 1.3 ±0.1 0.52 ±0.18 Mg 24(77%), 25(12%), 26(11%) 1.4 ±0.1 0.75 ±0.23 S 32 2.35 ±0.15 0.38±0.1 Cr 52(84%), 53(10%), 50(4%), 54(2%) 1.4 ±0.1 1.2 ±0.4 Fe 56(92%), 54 (6%), 0.8 ±0.1 1.8 ± 1.3 57(2%), 58(0.3%) 2.2 ±0.2 0.14 ±0.05 Cu 63(69%), 65(31%) 1.1 ±0.1 1.2 ±0.6 2.2 ±0.1 0.34 ±0.12 (b) Interpretation Tx/r Ele- (barns) Refer- ence Estir <r c (b nated arns) ment Experii nent Theory F 2.7 W32 1 .2 0.4 0.41 Mg 2.2 G20, A2 1 .5 0.5 (0 .4) 0.40 S 2.8 G20, A2 2 0.2 0.19 (2) Medium Nuclei. In the scattering of fast neutrons by medium nuclei, the product nucleus can be left in any one of many (energetically) available excited states; correspondingly, many groups of inelastically scattered neutrons will be emitted by the compound nucleus. The con- siderations applied in the previous section for a single level can easily be generalized to the case of many levels. Let c; be the cross section for the excitation, through inelastic scattering, of the z'th level of the target nucleus (excitation energy Z>,) : <n = (42b) Sec. 2C] Interaction with Nuclei 303 The partial width I\- is obtained by the substitution of the subscript i for 1 in Eqs. (38a) and (33a). Since, for intermediate nuclei and fast neutrons, neutron re-emission is usually predominant over all other processes, the total width is n r = r + r : +• • •+ i\ + • • •+ r n = £ r,- (28b) in which the first term is the width for elastic re-emission and the suc- ceeding terms correspond to inelastic scattering in which all possible levels (up to the highest for which D n < E) of the target nucleus are excited. As in the case of one-level excitation, the value and energy depend- ence of a specific I\- are determined by the angular momentum proper- ties of the emitted neutrons which, in turn, derive from the spins and parities of the levels involved. Since, for medium compound nuclei formed by the capture of fast neutrons, many levels involving many angular momenta are likely to be excited, decay to a given product nucleus state will be possible through the emission of neutrons of a number of different angular momenta. In the ensuing competition, 1 = emission when possible, will usually predominate. However, for some levels, I = emission will be impossible, and these will usually be less strongly excited. For a complete description it is, of course, neces- sary to take into account all the possible I values of the emitted neutrons. Nevertheless, for purposes of illustration it is of interest to consider the energy dependence of the inelastic scattering cross sections under the assumption of <S-wave scattering only. The values of the first few <Ti vs. E are shown in Fig. 14; in addition to the assumption of $-wave scattering, we have also assumed uniform level spacing for the product nucleus, i.e., Z>i = D, D t = iD. The uppermost curve, labeled a oc , is the relative cross section for capture elastic scattering. At a given value of the incident neutron energy E, the total cross section for in- elastic scattering is n "in = z2 c i ~ °c ~ "oc (68') i=l The data on the inelastic scattering of fast neutrons by intermediate nuclei are meagre. Barschall, Battat, Bright, Graves, Jorgensen and Manley (B8) have measured the energy distribution of 3.0 and 1.5 Mev neutrons scattered by iron, using, as an energy-sensitive detector, pro- ton recoil proportional counters with different "bias" values. Although the resolution of these measurements was rather crude, the results can be satisfactorily understood in terms of the theory outlined above and 304 The Neutron [Pt. VII the (three) known levels of Fe 56 of excitation energy less than 3 Mev (F9). The results of Grace et al. (G27), shown in Table 7a, are in fair agreement with those of Barschall et al. for iron. (3) Heavy Nuclei. In the scattering of fast neutrons by heavy nuclei a large number of energy levels of the target nucleus are available for excitation. Separate groups of inelastically scattered neutrons will, in general, not be observed, both because of the limited resolving power of Fig. 14. Cross sections, <r,-, for the excitation of the low-lying levels of the target nucleus as a function of the incident neutron energy, E. The curves are based on the assumptions: (1) constant level spacing; (2) S scattering only; (3) I\ = C{E — Ei) Vi , with C constant over all levels. all fast-neutron detectors and because the separation of levels in the product nucleus is usually smaller than the spread in energy of avail- able fast-neutron sources. The inelastically scattered neutrons will appear to have a continuous spectrum (except, possibly, for the elasti- cally scattered and a few adjacent high-energy groups) ranging from zero to the incident energy E. Because of the large number of levels involved, it is no longer fruitful to attempt to interpret such experiments in terms of a theory involving the properties of individual levels. Instead, it is possible to apply a statistical theory, as developed by Weisskopf (W16, B34). The statisti- cal theory predicts an energy distribution of the inelastically scattered neutrons which is of the Maxwellian form da(e,E) ^<To— 2 e- E/r dz s Ji2 (69) Sec. 2C] Interaction with Nuclei 305 In the above, da(e,E) is the cross section for the scattering of a neutron of initial energy E into the energy between e and s + de. The constant T, referred to as the nuclear temperature, is a measure of the excitation of the -product nucleus after the emission of the inelastically scattered neutron. Strictly speaking, it is not energy independent but rather a function of the excitation energy of the residual nucleus, E — e. How- ever, for most cases of interest, T « E, and the major part of the spec- trum of inelastically scattered neutrons is in the energy region s <JC E; for this part of the spectrum, T may be regarded as essentially con- stant, and roughly corresponding to the full possible excitation energy of the product nucleus. However, for the high-energy portion of the spectrum, for which e ~ E, Eq. (69) is not a good approximation to the inelastically scattered neutron energy distribution. The energy dependence of Eq. (69) can be understood in terms of two opposing factors in the competition between the various possible modes of de-excitation of the compound nucleus through neutron re-emission: (1) The energy dependence of the neutron scattering width favors the emission of high-energy neutrons. This effect is responsible for the first factor e in Eq. (69) when proper account is taken of the emission of neu- trons in all possible angular momentum states. (If neutrons were emit- ted only in the I = state, the factor would be s H .) (2) The number of available levels of the product nucleus increases rapidly with the exci- tation energy, thus favoring the emission of low-energy neutrons. This effect leads to the exponential factor in Eq. (69). The competition between the two factors results in a maximum, in the scattered neutron energy distribution, at an energy intermediate between and E. T, the temperature of the product nucleus, is a measure of its level density at the excitation energy remaining after the emission of the inelastically scattered neutron. From statistical- mechanical considerations, it can be shown that (B34) 1- d = [In u(E')] (70) T(E') dE' K J where w(E') is the nuclear level density at the excitation energy E' = E — s. For an exponential energy dependence of the nuclear level density, as given by Eq. (41), T(E') ~ (-J (70a) In heavy nuclei, the very rapid increase of nuclear level density with increasing excitation energy has the effect that the maximum of the 306 The Neutron [Pt. VII scattered neutron energy distribution is at relatively low energies; cor- respondingly, the excitation energy of the product nucleus is, for the major fraction of the inelastic scattering, at an energy close to the maxi- mum possible excitation energy, E' <~ E. [This rapid increase in level density corresponds to large values of the constant a (Table 4) and 2.4 2.0 1.6 1.2 0.8 i 0.4 w (a) E = 3 Mev ^- ■ 1 I , 1.0 1.6 E(Mev) - / ^v- (b).E=1.5 Mev I - ^ ~ 1 0.5 E(Mev) 1.0 Fig. 15. Energy distribution of inelastically scattered neutrons from wolfram. The experimental results of Barschall et al. (B8) are plotted as histograms. The smooth curves are derived from the statistical theory and Eq. (69) (F9). hence to small values of the nuclear temperature, T <K E.] Thus, over the largest part of the spectrum, the nuclear temperature can be closely approximated by T(E) 0" (70b) which is independent of the energy of the scattered neutrons. It is in this approximation that Eq. (69) is valid. The observations of Barschall et al. (B8), on the inelastic scattering of 1.5 and 3.0 Mev neutrons by wolfram (tungsten), can be interpreted in terms of the statistical theory and Eq. (69) (F9). The results of their measurements on the energy distribution of the inelastically scattered neutrons are plotted as histograms in Fig. 15, together with the predic- tions of the theory (smooth curves). The theoretical curves have been % (E,E t ) = C Jo Sec. 2C] Interaction with Nuclei 307 computed on the assumption of T = 0.5 and 0.35 Mev for E = 3.0 and 1.5 Mev, respectively, corresponding to a = 12 Mev -1 , in good agree- ment with Table 4. Table 8 is a collection of integral inelastic scattering cross sections and temperatures for medium and heavy nuclei. The measurements are represented by the cross section for the scattering of a neutron of incident energy E to an energy e < E t , dc(E,z) (69a) 'o and by the temperature corresponding to the emitted neutron distribu- tion. Only those measurements have been included for which the inci- dent neutron energy and the detector threshold are relatively well defined. Some of these are amenable to interpretation along the lines outlined above in the discussion of the results for wolfram. For most, the energy sensitivity of the threshold detector used is not sufficiently well known to permit more quantitative conclusions. However, these measurements do give an indication of the energy distribution of the inelastically scattered neutrons. In particular, they show that, also with regard to inelastic scattering, the magic number nuclei lead and bismuth behave in an anomalous fashion; their scatter- ing is much more similar to that of medium nuclei than to that of heavy nuclei. The importance of this observation is that it relates to the rela- tively low-lying level spacings of the target nuclei, which appear, from these observations, to be anomalously large for lead and bismuth. The previously mentioned evidence on the small capture cross sections and absence of slow-neutron resonances for these nuclei reflect a wide level spacing of the compound nucleus. In addition to the direct observation of inelastically scattered neutrons or the resulting gamma-radiation, there is another method of detecting inelastic scattering, applicable only to certain special nuclei. These are nuclei which have a metastable (long half-life) level at an energy less than that of the incident neutrons; when excited, the metastable state can be detected by the resultant radioactivity. The metastable state of In 115 (energy 340 kev, half-life 4.5 hr) was first observed by Gold- haber, Hill, and Szilard (G13) as resulting from the inelastic scattering of fast neutrons. A metastable state can be induced either by direct excitation (the product nucleus is left in the metastable state), or indirectly, through the excitation, by inelastic scattering, of a higher state of the product nucleus, which subsequently decays (by gamma-ray emission) to the metastable state. In the first case, Eqs. (42a) and (38a) determine the 308 The Neutron [Pt. VII TABLE 8 (a) Measured Values of Integral Inelastic Scattering Cross Sections, <n n {E,E t ) Ele- ment E (Mev) E t (Mev) Detec- tor t <Ti n (E,E t ) X (barns) Refer- ence <j c (barns) (Fig. 10) Be 2.5 2.5 y <0.014 G27 14 ~3 Al 0.16 ±0.07 P15 ~11 Cu 0.82 ±0.03 P15 0.79 B 14 ~3 Al 0.24 ±0.04 P15 ~11 Cu 0.69 ±0.10 P15 0.64 C 2.5 2.5 7 <0.006 B13 14 ~11 Cu 0.85 ±0.02 P15 0.74 F 2.5 2.5 7 0.52 ±0.18 G27 1.4 Mg 2.5 2.5 7 0.75 ±0.23 G27 1.10 ~2.5 cc ~0.6 L20 Al 14 ~3 Al 0.62 ±0.07 P15 ~11 Cu 1.06 ±0.05 P15 0.93 S 2.5 2.5 7 0.38±0.1 G27 0.90 Cr 2.5 2.5 7 1.2 ±0.4 G27 1.47 Fe 1.5 ~0.5 pc B8 ~0.9 pc 0.6 B8 1.58 2.5 2.5 7 1.9 ± 1.3 G27 1.52 3.0 ~0.75 pc 0.3 B8 ~1.50 pc 0.7 B8 ~2.25 pc 1.1 B8 1.50 14 ~2 p 0.78 ±0.03 P15 ~3 Al 1.21 ±0.03 P15 ~11 Cu 1.45 ±0.02 P15 1.34 Co 1.5 ~0.5 pc (0) B8 ~0.9 pc (0.2) B8 ~1.3 pc (0.8) B8 1.62 Ni 1.5 ~0.5 pc (0) B8 ~0.9 pc (0.1) B8 ~1.3 pc (0.6) B8 1.62 Sec. 2C] Interaction with Nuclei 309 TABLE 8 (Continued) (a) Measured Values of Integral Inelastic Scattering Cross Sections, <Ti n (E,E t ) Ele- ment E (Mev) E, (Mev) Detec- tor f a in (E,E t ) t (barns) Refer- ence (T c (barns) (Fig. 10) Cu 1.5 ~0.5 pc (0.3) B8 ~0.9 pc (0.6) B8 ~1.3 pc (0.9) B8 1.65 2.5 2.5 y 1.5 ±0.7 G27 1.62 3.0 ~0.75 pc (0.6) B8 ~1.50 pc (1.3) B8 ~2.25 pc (1.5) B8 1.61 Cd 14 ~2 p 1.14 ±0.04 P15 ~3 Al 1.66 ±0.07 P15 ~11 Cu 1.89 ±0.06 P15 1.85 Ta 1.5 ~0.5 pc (1.4) B8 ~0.9 pc (2.0) B8 ~1.3 pc (2.7) B8 2.57 W 1.5 ~0.5 pc 0.9 B8 ~0.9 pc 2.1 B8 2.57 3.0 ~0.75 pc 1.4 B8 ~1.50 pc 2.4 B8 ~2.25 pc 2.8 B8 2.63 Au 3.0 ~0.75 pc (2.1) B8 ~1.50 pc (2.8) B8 ~2.25 pc (3.0) B8 2.69 14 ~2 p 1.47±0.10 P15 ~3 Al 2.06 ±0.09 P15 ~11 Cu 2.51 ±0.04 P15 2.69 Pb 1.5 ~0.5 pc B8 ~0.9 pc 0.4 B8 2.81 2.5 ~1 U 0.55 S65 ~2.5 cc 1.3 ±0.5 D17 2.77 3.0 ~0.75 pc 0.7 B8 ~1.50 pc 1.2 B8 ~2.25 pc 1.6 B8 2.76 14 ~2 p 0.91 ±0.06 P15 ~3 Al 2.29 ±0.04 P15 —11 Cu 2.56 ±0.05 P15 12 PP <2.6 W21 2.77 14.5 ~3 Al 2.20 ±0.17 G6 ~11 Cu 2.29 ±0.12 G6 2.76 310 The Neutron [Pt. VII TABLE 8 (Continued) (a) Measured Values of Integral Inelastic Scattering Cross Sections, Oin(E,E t ) Ele- ment E (Mev) E t (Mev) Detec- tor f Oin(E,Et) % (barns) Refer- ence <r c (barns) (Fig. 10) Bi 2.5 ~1 U 0.64 S65 2.77 14 ~2 P 1.03 ±0.11 P15 ~3 Al 2.28 ±0.08 P15 ~11 Cu 2.56 ±0.05 P15 12 VP <3.3 W21 2.78 (b) Temperatures for Observed Distributions of Inelastically Scattered Neutrons (Eq. 69) Element E (Mev) Range of e (Mev) T{E) (Mev) Reference B 11 >4 2.3±0.3 G31§ 1-4 0.9±0.1 G31 Al 15 >1 1.1 ±0.1 S53 Si 10.6 >2 1.3±0.1 G31 Fe 1.5 <0.9 none || B8, F9 3.0 <2.25 none B8, F9 15 >1 0.6±0.1 S53 Co 10.5 >2 0.95 ±0.1 G31 Pd 14 >2 0.85±0.1 G31 W 1.5 <0.9 0.35 B8, F9 3.0 <2.25 0.50 B8, F9 Au 3.0 <2.25 (0.33) B8, F9 Hg 14.6 >2 0.8± 0.1 G31 Pb 1.5 <0.9 none B8, F9 2.5 >1 none D17, F9 3.0 <2.25 none B8, F9 4.3 >1 none M2 Sec. 2C] Interaction with Nuclei 311 TABLE 8 {Continued) (b) Temperatures for Observed Distributions of Inelastically Scattered Neutrons {Eq. 69) Element E (Mev) Range of e (Mev) T{E) (Mev) Reference 14 1-3 —0.8 W21 14.3 >2 0.78±0.1 G31 15 >1 0.7±0.1 S53 Ki 4.3 >1 none M2 14 1-5 —0.9 W21 f The symbols have the following meanings : 7 = direct detection of gamma-rays from the product nucleus P = T 31 {n,p) reaction Al = Al 27 (n,p) reaction Cu = Cu 63 (ra,2w.) reaction pc = detection in a biased proton recoil proportional counter pp = detection by proton recoils in a photographic emulsion cc = detection by proton recoils in a cloud chamber U = detection by U 238 fission in an ionization chamber t Values in parentheses have not been corrected by the authors (B8) for effects of multiple scattering in the target. § Reference (G31) is to measurements of Gugelot on neutron distributions from {p,n) reactions. In these cases, the element given is that of the product nucleus and the energy (second column) is the maximum neutron energy, i.e., the proton kinetic energy plus the reaction Q value. || "None" means that the inelastic scattering cannot be treated by the statis- tical theory. Instead, individual levels must be considered. behavior of the cross section for the excitation of the metastable level; in the second case, the partial cross section for the production of the metastable state through the excitation of the ith level is given by the product of Eq. (42a) and a factor representing the relative probability for the decay of the ith level to the metastable state. The excitation function (total cross section vs. neutron energy) is the sum of the partial cross sections. It will exhibit discontinuities at the energies correspond- ing to the inception of excitation of those product nucleus levels which have an appreciable probability of decaying to the metastable level. The cross section for the excitation of the metastable state of In 115 by neutrons of energy up to —4 Mev has been measured by Cohen 312 The Neutron [Pt. VII (CI 7). The resolution of his measurements was too crude to detect the effects of individual levels of the product nucleus. More recently Ebel (El) has measured the excitation function for In 115 * with good resolu- tion. He observes a threshold at 600 kev (no direct excitation of the metastable level) and discontinuities corresponding to additional prod- uct nucleus levels at 960 and 1370 kev. Ebel has also measured the cross section for the excitation of the 540-kev metastable level (half-life 7 sec) of Au 197 . This level can be directly excited, and also excited through levels at 1.14 and 1.44 Mev. Since the metastable state is characterized by a large difference in spin from the ground state, the levels of the product nucleus which are involved in its excitation are, perforce, those whose spins differ appre- ciably from that of the ground state. The shape and magnitude of the cross section for excitation of the metastable state can be used to deter- mine (within limits) the spins and parities of the states involved in its excitation (El). II. Diffraction or Shadow (Non-Capture) Elastic Scattering In the previous discussions, we have considered the total cross section as consisting of two parts : (1) that portion which leads to the formation of a compound nucleus, <r c , and (2) that portion corresponding to proc- esses in which the incident neutron merely changes its direction, with- out ever effecting a change in the quantum state of the target nucleus; l this elastic scattering, <r e i (see Eq. 68), is variously referred to as diffrac- tion or shadow scattering. For slow and intermediate neutrons (X > R) capture and diffraction scattering are intimately connected because of the interference between resonance and potential scattering. For fast neutrons (X < R) the two processes separate in a natural fashion. In the first place, the position (extent) of the neutron is relatively well defined. Those neutrons which strike the nucleus have a high probability of penetrating its surface, whereupon they are captured into a compound nucleus. While a cer- tain fraction (relatively large for light nuclei, small for medium and heavy nuclei) may be re-emitted with the full energy, these (capture) elastically scattered neutrons can, except for interference effects in the 1 The target nucleus, initially at rest, will of course receive momentum and kinetic energy as a result of the scattering, and the neutron will, correspondingly, lose energy. However, by the term "quantum state" we refer to the state of internal motion of the target nucleus, which is unaffected unless the incident neutron penetrates the nuclear surface. Sec. 2C] Interaction with Nuclei 313 neighborhood of a resonance, be separated from the diffraction scattered neutrons by virtue of their different angular distribution. Shadow scattering, on the other hand, results from- the diffraction of those neutrons which pass close by, but not into, the nucleus. Thus, despite the fact that we are considering the energy range for which X < R, in which neutrons may be expected to exhibit a minimum of wave properties, it is precisely the wave nature of the incident neutron beam which leads to the phenomenon of shadow scattering. Further- more, while the angular distribution of the diffraction elastically scat- tered neutrons depends on the value of the nuclear radius (specifically, on kR = R/X), the nature of the diffraction scattering process is inde- pendent of the nuclear atomic number. To a good approximation, shadow scattering is analogous to the diffraction of a plane wave (say light) by a spherical obstacle, for wave- lengths small compared to the size of the obstacle. The scattered neu- trons are confined to within a relatively small angle, 0o « ^ (71) An expression for the angular distribution of the scattered neutrons was first derived by Placzek and Bethe (PI 7) on the basis of the optical analogy. Recently, Hauser and Feshbach (H44) have derived a more accurate formula, -~^ = ' cot 2 (-){Ji[fc(fl + X) sine]} 2 (72) dQ. 4 \2/ for the cross section per unit solid angle for scattering of neutrons into the angle 9; J t is the Bessel function of the first kind. The cross section has a maximum at = 0, falls to zero at k{R + X) sin 6 = 3.83, and then goes through a series of subsidiary maxima and minima. For very small wavelengths, X <K R, the cross section for shadow (elastic) scattering is "el = fdv(0) « r(B + X) 2 (73) At this extreme, a t « 2ir(R + X) 2 (Eq. 67), so that diffraction elastic scattering accounts for approximately half of the total cross section for fast, very fast, and ultra fast neutrons. The experiments of Amaldi, Bocciarelli, Cacciapuoti, and Trabacchi (A21) on the angular distribution of the elastic scattering of 14-Mev 314 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 The Neutron [Pt. Vii v?= 8.5 a(8; vs. 6 _| E-- =14 W ev R='t .5 I ^w \ to R = ( 5.5 R = l >.5 R=A ??~~~ X X x X X i? = 3.5 10 15 20 25 30 35 40 45 50 55 60 65 Fig. 16. Angular distribution in diffraction elastic scattering of 14-Mev neutrons for a number of nuclear radii (given in units of 10~ 13 cm). The curves are accord- ing to the theory of Hauser and Feshbach (H44). The crosses are from the measure- ments of Amaldi and co-workers on lead (A21). Sec. 2C] Interaction with Nuclei 315 neutrons by lead are in excellent agreement with the theory. In addi- tion to the general shape of the scattered neutron distribution, they have observed subsidiary maxima at approximately the expected angles. The theoretical predictions for E = 14 Mev and a number of values of the nuclear radius are shown in Fig. 16. Also plotted are the experimen- tal points of Amaldi and co-workers (A21). III. Angular Distribution of Scattered Neutrons; the Transport Cross Section o> The angular distribution of the scattered neutrons, with respect to the direction of the incident neutrons, is usually described in terms of a differential scattering cross section, <r sc (0), the cross section for scatter- ing into unit solid angle at the angle 6, in the laboratory system, f«c = f<Tac(6) da (73a) Deviations from spherical symmetry result from a number of possible causes, some of which have previously been discussed. Since the angu- lar distributions are determined by the nature of the scattering process, measurement of o- sc (0) can sometimes lead to useful nuclear information. The angular distribution of scattered neutrons is also important in determining the rate of diffusion of neutrons through matter (to be dis- cussed in a subsequent section). In such problems, the important quan- tity is the transport cross section, atr = <rt- J <T, C (0) cos 6 da (74) a tr determines the rate at which the neutron loses its forward momentum (or the memory of its original direction). Under certain special assump- tions concerning the nature of the scattering process simple expressions for <Tt r can be derived. (1) Light Nuclei. In the scattering of fast neutrons by light nuclei, elastic scattering is the most important process. Furthermore, the scattering is (except at the highest energies) mainly £-wave scattering, i.e., spherically symmetrical in the center-of-mass coordinate system (c.m.s.). Neglecting all processes but $-wave elastic scattering, the angular distribution of the scattered neutrons is still not symmetrical in the laboratory system (l.s.), owing to the forward motion of the center of mass. 316 The Neutron [Pt. VII Thus, for a neutron, scattered through the angle <p, in the c.m.s., by a nucleus of atomic weight A, the angle in the l.s. is given by (1 + A cos <p) (A 2 + 1 + 24 cos?)' cos 6 = , A 2 , , , »„ — VA W E (75a) The energy in the l.s. of the elastically scattered neutron, of initial energy E, is r(A 2 + l + 2Acos**r E = L {A + l) 2 the difference (E — E') going to the recoil nucleus. The energy of the scattered neutron varies between a maximum, E mB ,^' = E, for <p = 8 = 0, and a minimum, _ -(A - iy 2 Er, ■'-[ (A + 1). # b a E (75b) for ? = 180°. For <S-wave scattering (in the c.m.s.) the scattered neu- trons are distributed uniformly in energy between E and aE. Corre- spondingly, under the above assumptions, 2 (Ttr = Ot ~ — "el (74a) oA. (2) Medium and Heavy Nuclei. It is evident from Eq. (74a) that the effect of the motion of the c.m.s. can be neglected for medium and heavy nuclei. On the other hand, elastic scattering is no longer spheri- cally symmetrical, owing to diffraction effects. Most of the capture leads to inelastic scattering, and the resulting neutrons are not necessarily symmetrically distributed, especially if only a few levels are involved (medium nuclei). However, for most purposes, it is reasonable to treat the inelastic scattering as though it were spherically symmetrical, and to substitute a e i for u sc in Eq. (74). With this assumption the theory of Feshbach and Weisskopf (F28) can be used to predict the energy de- pendence of atr- The results are shown in Fig. 17, together with the experimental values of Barschall et al. (B8). (d) Very Fast Neutrons. So far as neutron scattering is concerned, the very fast-neutron energy range does not differ in any significant re- spect from the fast-neutron range. Most of the preceding discussion applies equally well to very fast neutrons; indeed, many of the experi- mental results quoted were for neutrons of energy > 10 Mev. However, in the extension of the previous considerations into the very fast-neutron region, the following points must be kept in mind: (1) Since many levels of the target nucleus can be excited by very fast neutrons, except for a Sec. 2C] Interaction witrfNuclei 317 few of the lightest nuclei, the inelastic scattering must be treated accord- ing to the statistical theory. (2) Because of the large excitation energy, the compound nucleus has, in addition to neutron re-emission, other possible modes of decay. Thus, for light and some medium nuclei, (n,p), (n,a), etc., reactions compete favorably with neutron re-emission. For medium and heavy nuclei, these reactions are also possible, but the {n,2ri) reaction is the most probable competing process. Although we postpone the detailed discussion of this reaction, it is important to note N\^o= > + Fe o W A Pb Xo=8 \N . +- Ir'-^^^l ■ i \ A ' : -=:==. i i i i i i I i i i i i i i i i Fig. 17. a tr vs. x = kR for nuclei of X = KoB = 5 and 8; according to Feshbach and Weisskopf (F28). The points, for iron, wolfram, and lead, are from the measure- ments of Barschall et al. (B8). that its presence has a strong effect on the observed energy distribution of the emerging neutrons. Experiments designed to measure the energy distribution of inelastically scattered neutrons must, in the very fast- neutron range, take into account the deviations from Eq. (69) brought about by the presence of the (n,2n) reaction (S53). The upper end of the very fast-neutron range is characterized, for all nuclei, by X « R. Thus at these energies <x t ~ 2x(fi + X) 2 and c c « cr e i » tt{R + X) 2 . (e) Ultrafast Neutrons. In the ultrafast-energy region, the character of neutron reactions undergoes a gradual change. The simple picture of neutron capture into a compound nucleus, as developed in the previ- ous sections, is no longer applicable. Instead, as pointed out by Serber (S21), the nucleons in the nucleus must be treated as relatively inde- pendent particles, since their energy of internal motion and of binding is smaller than the energy of the bombarding neutron. Furthermore, owing to the rapid decrease of the primary (n-p) and (n-n) cross sections with increasing neutron energy — roughly as 1/E in the energy range 50 318 The Neutron [Pt. VII to 250 Mev — the incident neutron has an appreciable probability of passing through the nucleus without undergoing any collisions, i.e., the nucleus is partially transparent to ultrafast neutrons, the more so the lighter the nucleus. Under these circumstances the probability of a nuclear reaction can be described in terms of an absorption coefficient (reciprocal mean free path) of nuclear matter for neutrons, 3Ao- * = ^ Cm (76) where Za{n-p) + (A - Z)a(n-n) a = - (76a) A [In Eq. (76a) a(n-p) and <r(n-n) are appropriate averages over the spec- trum of relative energies of the incident neutron and the nucleons in the nucleus, appropriately reduced to take into account the effect, due to the exclusion principle, that not all energies are available to the recoiling nucleons.] Defining <r„ as the cross section for a collision of the incident neutron inside the nucleus, integration over a spherical nucleus of radius R yields ,1 1 - (1 + 2 K R)e- 2KR } The nuclear transparency also influences the diffraction elastic scat- tering of ultrafast neutrons, both in the magnitude of <?d — <r e i and in the angular distribution of the scattered neutrons. These effects have been considered in the calculations of Fernbach, Serber, and Taylor (F25). They have derived the diffraction scattering of a partially transparent sphere, characterized by the propagation vector k + fci = [2m(E + V)] 1A /h. (V is the effective nucleon potential inside the nucleus; k = 1/X is the incident neutron wave number.) The results of these calculations, for the special case kyJK = 1.5, are shown in Fig. 18, in which a^, c a , and cr t = a a + <?d are plotted against kR. For large values of kR (complete opacity) <r a » ad — > kR 2 . Fernbach, Serber, and Taylor have compared their calculated values of (T t with the measurements of Cook, McMillan, Peterson, and Sewell (C25), for E « 85 Mev. They find excellent agreement, assuming R = 1.37A M X 10~ 13 cm, k = 2.2 X 10 12 cm -1 , and k t = 3.3 X 10 12 cm -1 (V = 30.8 Mev). The measurements of <r t at E « 95 Mev, by DeJuren and Knable (D9) are also in good agreement with theory. These investi- gators have, in addition, determined a a for two elements (carbon and copper), by a "poor geometry" attenuation measurement, and obtained values in good agreement with the theory. Sec. 2C] Interaction with Nuclei 319 The angular distributions of 84-Mev neutrons elastically scattered by aluminum, copper, and lead have been measured by Bratenahl, Fern- bach, Hildebrand, Leith, and Moyer (B59). The results are shown in Fig. 19 in which the dotted curves are the predictions of the theory of Fernbach, Serber, and Taylor. Measurements of the total cross sections of a number of elements have been carried out at 280 Mev by Fox, Leith, Wouters, and MacKenzie (F41), and at 270 Mev by DeJuren (D10), who also obtained lower Fig. 18. Absorption, diffraction, and total cross sections as functions of the nuclear radius, measured in mean free paths for the incident neutron. The curves are for ki/i< = 1.5, corresponding to neutrons of E « 90 Mev (F25). limits for <r of carbon, copper, and lead by "poor geometry" attenuation measurements. The above-mentioned measurements of a t are included in the compilation of Adair (A2). Cross sections of some representative elements have been measured at a number of intermediate energies — between 110 and 240 Mev— by DeJuren and Moyer (D12). A rather interesting common feature of all the observed <x t vs. E curves is their comparative independence on energy above <~160 Mev. Another point of interest is the relative constancy of the ratio <r t (270 Mev)/<r* (95 Mev) ; its value is 0.57 between beryllium and tin and then rises slowly to 0.67 for uranium. The last feature is difficult to reconcile with the partially transparent nucleus model, which predicts considerably greater nuclear opacities for the heavier nuclei. 1 Indeed, for the 270-Mev data, it is only possible 1 This peculiar behavior of <r ( for neutron energies > 200 Mev may possibly be associated with the inception of meson production. While the threshold for pion creation is ~275 Mev in a nucleon-nucleon collision, meson production is observed at considerably lower energies for the bombardment of nuclei by nucleons. This is due to the internal motion of the nucleons in the nucleus. 320 The Neutron [Pt. VII o 1 - 1 1 ' Al — r — ' — i — ' I ' I ' i ' 1 1, | o Carbon detector measurement ~^ 1 X* 1 . 1,1,1,1 .~-T-^- 24° 20 .... , , 1 Cu r-T — ' "7" i 1 I 1 ' 1 » 18 ~ c a is 1 " o ,4 S 14 to - ii c "^0 "" a & \ VI 1 1° - - \ \ a \ \ 4 \ 2 ' . I.I, r« J. l-a— 0° 12° 16° 20° 80 ft 70 .2 % 60 w at O 50 O 1 40 0) S 30 — i — i — i — i 1 — i — r— i — - Pb ,,.. , _, , - Ntf •- \ \ \ \ 1 i 1 ^-tf-^-e-T— T-T---J ' 4° 8° 12° 16° 20° 24° Neutron scattering angle Fig. 19. Differentia] cross sections for elastic scattering of 84-Mev neutrons by aluminum, copper, and lead, due to Bratenahl, Fernbach, Hildebrand, Leith, and Moyer (B59). The curves are the predictions of the theory of Fernbach, Serber, and Taylor (F25). Sec. 2C] Interaction with Nuclei 321 "to obtain a reasonable fit . . . [if] the potential change experienced by the bombarding neutron when entering a nucleus [is] dropped to zero" (D10). 2. Radiative Capture (n,-y). The excited compound nucleus, produced by neutron capture, can decay to its ground state through the emission of one or more gamma-rays. The (n,y) reaction competes with neutron re-emission (always possible) and with any other mode of decay which is energetically permitted. The relative probability of the (n,y) reac- tion is determined by the value of the gamma-ray width, r r , as com- pared to the widths for all other possible modes of decay: *(n,y) = - 1 (42') At the excitations corresponding to neutron binding energies, the emission of a gamma-ray takes place in a time of ~10~ 14 sec; corre- spondingly (Eq. 27) r T ~ 0.1 ev. This value is relatively independent of the atomic weight of the nucleus involved and of the incident neutron energy (at least for slow and intermediate neutrons), although con- siderable fluctuations, from nucleus to nucleus and from level to level in the same compound nucleus, are to be expected. The above lifetime refers, of course, only to the emission of the first gamma-ray, after which the compound nucleus is usually committed to radiative decay, not having sufficient residual energy to decay by any other mode. In some cases, successive gamma-ray emission leads to a metastable (iso- meric) state of the compound nucleus. Usually, however, the gamma-ray emission leads, either directly or through a cascading descent, to the ground state of the compound nucleus. The study of the energy distribution of the capture gamma-radiation is of considerable interest, since it yields information on the level struc- ture of the compound nucleus, on the nature of the gamma-ray emission process, and on neutron binding energies. Furthermore, knowledge of the capture gamma-ray spectrum is of practical importance in the prob- lem of shielding neutron chain reactors. However, since further discus- sion of these problems at this point would lead us too far afield, we shall be content merely to point out that the observation of the capture gamma-radiation is one means of detecting the (n,y) reaction. In many (n,y) processes the product nucleus is a stable isotope of the target, e.g., Cd 113 (n, 7 )Cd 114 ; in such cases, the capture gamma-rays are the only reaction products. However, for most nuclei, radiative capture leads to an unstable (radioactive) product nucleus. The reac- 322 The Neutron [Pt. VII tion can then be detected and identified by observing the resulting radio- activity. Since these radioactive nuclei are most likely to have an excess of neutrons, they usually decay by negative beta-ray emission. There are a few cases, e.g., Cu 63 (n,7)Cu 64 , in which the product nucleus can decay by positron emission or K capture. Most of the factors which determine the energy dependence of <r(n,y) have previously been described, both in the general discussion of the cross section for compound nucleus formation, a c , and in Section 2C1 on neutron scattering. In the following, we summarize the main fea- tures of the radiative capture process. (a) Slow Neutrons. The value of <r(n,y) at a given slow-neutron en- ergy is determined by two factors: (1) The position and characteristics of the closest resonance or, if the energy is far from any one resonance (| E — E„ \ ~2> r,-), the positions of the closest resonances and their possible interference effects at the energy under consideration. The last aspect has been discussed by Wigner and co-workers (W29, W30, T5). The behavior of <r{n,y) in the vicinity of a resonance has been considered in detail (Section 2B3, espe- cially Eqs. 52, 56, 58, and 59). (2) The relative value of the gamma-ray width, r 7 , to the total width, r = T y + V n + T a + • • • ; in all but a few light nuclei only the first two terms, in the above expression for r, differ from zero. Since T y is, as per the previous discussion, essentially fixed, the competition between radiative capture and neutron scattering is primarily determined by the value of r n , which, in turn, is a function of the neutron energy and the compound nucleus level spacing, according to Eqs. (37) and (38). From these considerations it is clear that the importance and the character of the slow neutron (n,y) cross section depend on the atomic weight of the target nucleus. In light, medium, and some heavy nuclei, levels are widely spaced, T n >?> T y , and radiative capture plays a minor role. In most heavy nuclei, on the other hand, levels are closely spaced, r 7 2> r„, and radiative capture is the predominant resonance reaction. The properties of a number of slow-neutron resonances are summarized in Table 9. The values of the resonance energy and level widths are given, as well as the deduced values of the level spacings, derived from Eq. (37a), for comparison with the observed (average) spacing between levels. Of particular interest in the study and use (to produce artificially radioactive nuclei, for instance) of radiative capture is the so-called thermal neutron capture cross section, <r t h.{n,y). This cross section has been variously defined, sometimes in a not very precise fashion. We Sec. 2C] Interaction with Nuclei 323 TABLE 9 Properties of Some Slow-Neutron Resonances — Mainly Capture f Target Nucleus E r (ev) I\ (ev) t /r„(10- 3 ev)§ D* (ev) || Dobs (ev) 17CF -75 0.3 1.64 X 10 s 2,200 26 Mn 8B 345 0.2 12 X 10 3 7,700 10,000 27 Co 59 123 0.3 2.4 X 10 3 2,800 10,000 3oZn 520 5-25 X 10 s 2-8,000 >500 3iGa< 69 ' 71 > 98 (0.3) 130 90 <100 32Ge 95 3-900 2-600 33AS" 46 0.05 110 120 50 35 Br«5,87) 36 (0.1) 35 86 30 42M0 86 46 (0.1) 450 1,000 50 4BRh 103 1.26 0.20 0.45 6 4 6 Pd 108 25 0.14 ~40 ~60 >10 47Ag 107 15.9 ~0.11 ~5 ~10 ~20 4rAg™ 5.17 0.16 8.2 34 —20 4 8 Cd 1W 0.176 0.115 0.46 10 ~25 49 In 115 1.45 3.86 9 0.08 (0.08) (0.08) 1.2 0.2 0.5 14 2 5 6 6 6 61 g b (121, 12 3) 5.8 15 (0.1) (0.1) ~0.3 ~2 ~2 ~8 7 7 B2 Te 123 2.2 (0.1) 8 70 ~1 53I 127 19.4 0.45 0.38 1.2 15 6 2 Sm» 9 0.096 0.074 0.31 13 2 324 The Neutron [Pt. VII TABLE 9 {Continued) Propebties of Some Slow-Neutron Resonances — Mainly Capture f Target Nucleus E r (ev) T y (ev) J /r„(10- 3 ev)§ D* (ev) || Dobs (ev) 63 Eu 161 -0.011 0.081 0.004 0.5 ~3 6 3 Eu lB3 0.47 0.20 0.9 18 ~3 64 Gd 167 0.028 0.12 0.4 30 66 Dy< 161 - 163) 1.74 (0.1) 3 30 ~3 7 2 Hf m 1.08 2.34 0.12 0.16 0.9 2.8 12 26 2 2 72 Hf 178 7.6 (0.1) 56 150 7 3 Ta 181 6.1 10.3 13.6 20 0.2 (0.1) (0.1) 0.0007 0.85 0.45 0.7 0.04 3.8 1.8 2.2 5 5 5 5 74W 182 4.15 0.07 1.1 4 ~20 74 W 183 7.8 (0.1) 1.0 5.3 20 74 W(184) ~200 (0.1) ~400 ~200 ~100 74W 186 19.25 0.15 250 400 ~150 76 Re 185 4.4 11 (0.1) 0.3 3.5 2 15 5 5 76 Re 187 2.15 5.9 7.2 0.14 0.43 0.09 0.40 4.2 0.5 2.1 3 3 3 76 0s< 189 > 6.5 8.8 (0.1) (0.1) 1.6 7.5 9 36 6 6 77 I r (Wl. W3) 0.64 0.10 0.2 4 2 77 Ir 193 1.27 ~0.15 0.4 5 2 Sec. 2C] Interaction with Nuclei 325 TABLE 9 (.Continued) Properties op Some Slow-Neutron Resonances — Mainly Capture f Target Nucleus E T (ev) T y (ev) i /r B (io- 3 ev)§ D* (ev) |! Dobs (ev) 78 Pt< 196 > 11.5 18.2 (0.1) (0.1) 6.5 5.5 27 19 10 10 79 Au I9r 4.87 0.15 21 136 ~50 8oHg (199,201) -2.0 35.5 (0.1) (0.1) 25 80 250 190 ~15 ~15 92U 238 ~11 ~0.20 ~8.6 ~20 t This table represents a complete re-evaluation of the data available in the open literature up to the summer of 1952. The measurements, upon which the constants are based, are so numerous that we have not attempted to include references to them in the table. Instead, the reader is referred to previous com- pilations, of which the most complete are those of Blatt and Weisskopf (B34), of Wigner (W31), and of Teichmann and Wigner [Phys. Rev., 87, 123 (1952)]. Special attention is called to the most recent neutron cross section compilation prepared by the AEC Neutron Cross Section Advisory Group and issued by the U. S. Department of Commerce, Office of Technical Services, as document AECU-2040 (May 15, 1952). We gratefully acknowledge private communica- tions of unpublished results by the Harwell time-of-flight velocity selector group (M26) and by the Argonne fast chopper group (whose members include L. M. Bollinger, R. R. Palmer, and S. P. Harris). Other data, pertaining to scattering resonances, can be found in Table 6. % The values enclosed in parentheses have been assumed in order to compute /T„ from the measured value of <r T 2 . § The numbers quoted are at the resonance energy, except for the negative energy resonances, where the neutron width is for the energy E = \E r \. || D* is computed from Eq. (37a): D* = TrK T n /2k. When the statistical factor / is not known, it is assumed to be 1 for even-even target nuclei and J for odd-A target nuclei. 326 The Neutron [Pt. VII shall define it as the average, over a neutron flux with a Maxwellian energy distribution (Eq. 3), of the slow neutron (n,y) cross section, ,E' a(n,y) d4(E,E ) <rth(n,y) = ^ (78) d<f>(E,E ) The cut-off energy, E', is chosen to be sufficiently large so that only a very small fraction of the Maxwell distribution is at energies E > E'. (Most frequently, the cadmium cut-off, E' ~ 0.3-0.5 ev is used; for E = "Fir ev > less than 10 -4 of the neutrons have E > W.) The cut-off is introduced for purely practical reasons, since (1) the cross sections above ~1 ev, which frequently show many resonances, are not well known, and (2) most methods for producing thermal neutrons give spectra which, above ~1 ev, have a 1/E "tail" superimposed on the Maxwell distribution. Unless otherwise specified, thermal neutron cross sections are taken to correspond to the Maxwell distribution at a temperature of ~300°K (E a = -jV ev, v Q = 2.2 X 10 5 cm/sec). If <r(n,y) is a known function of E, a t h can be computed in a straight- forward fashion. For nuclei in which the first resonance is relatively far from thermal energies, the (n,y) cross section obeys the l/v law, (59a) and (79) The thermal neutron (n,y) cross section may deviate from the l/v law in a spectacular fashion if a resonance happens to fall in the thermal re- gion. The cross section of cadmium, shown in Fig. 20a, is a case in point. However, even if the closest resonance does not fall in the ther- mal region, it may still cause serious deviations from a l/v behavior for thermal neutrons. The cross sections of iridium (Fig. 7d) and of mer- cury (Fig. 20b) show, respectively, the effects of a close positive-energy (E r > 0) and negative-energy (E r < 0) resonance. Because of such effects, considerable care must be taken in interpreting measurements of (x t h in terms of tr . Table 10 is a collection of thermal neutron (n,y) capture cross sections. cr(jl,y) = V •rofof v 2 e-" y '" ? dv •Jo ^ (fth r^' - (TO I v 3 e -v'W dv 2 Sec. 2C] Interaction with Nuclei 327 10,000 7 4 1000 <= 100 Is 7 1 Cd / / \ Symbol en Reference No. Figure in Legend • 1 o 2 \ \ 1 \ \ \ V \ \ o 0.01 0.1 1.0 -E n (ev) (1) Rainwater, Havens, Wu, and Dunning, Phys. Rev., 71, 65 (1947). (2) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). Also see W. H. Zinn, Phys. Rev., 71, 575 (1947); Sawyer, Wollan, Bernstein, and Peterson, Phys. Rev., 72, 109 (1947). (a) 2 ~ 1 E ioo oo**. I n — I Hg P V °^oo '°*a 1 2 ♦ 3 ■\o N ^ •v, ^ V, '<*■ » -VI • . • . 2 4 2 4 I 4 2 4 4 0.01 1.0 100 •BnCev) 1000 (1) W. W. Havens, Jr., and L. J. Rainwater, Phys. Rev., 70, 154 (1946). (2) L. B. Borst el al. (unpub- lished). (3) C. T. Hibdon and C. O. Muehlhause, Phys. Rev., 76, 100 (1949). (b) Fig. 20. Slow-neutron cross sections showing the effects of a resonance at or near thermal energies; from Adair (A2). (a) Cadmium, E r = 0.176 ev. Note the sharp cut-off at ~0.3-0.5 ev. (b) Mercury, E r = —2.0 ev, showing the influence of a close negative-energy resonance. 328 The Neutron [Pt. VII TABLE 10 Radiative Capture Cboss Sections fob Slow Neutrons Element At <r(n,y) t (barns) Neutron Energy § E r for Closest || Observed Resonance (ev) References 1[ iH 1 0.313 ±0.013 »0 R20 2 0.00032 P R20 3L1 7 0.033 th W10, H68 4 Be 9 0.0085 to 620,000 R20 6 B 11 <0.05 th 430,000 W10 6 C 0.0045 th ~4, 000, 000 R20 13 ~0.1 th W10 14 <200 th W10 7 N 15 0.000024 th tt 8 0.00028 ±0.00022 P 440,000 R20 18 0.00022 th W10, S22 9F 19 0.0094 ±0.0015 P 32,000 R20, S24 loNe <2.8 P tt u Na 23 0.50 th,P 3,000 R20, H42, P23, C21 12M2 0.057 P 230,000 W10, H42, P23, C21 24 0.033 P tt 25 0.27 P tt 26 0.05 ih,P S24, XX isAl 27 0.22 P 2,300 W10, H42, P23, C21 l«Si 0.13 P.th 600,000 W10, H42, P23, T14, C21 28 0.08 P » 29 0.27 P tt 30 0.12 th.P S24, tt 15P 31 0.17 P W10, P23, C21 ieS 0.49 ±0.05 »o 111,000 R20 34 0.26 th W10, S24 36 0.14 th W10 17CI 32 P W10, H42, P23, C21 35 42 th -75 H59 37 0.6 th 1,800 W10, S24 1»A 0.77 «o >1,000 R20 40 1.2 P W10 isK 2.0 P 65,000 W10, H42, C21, P23 39 ~3 th W10, H31 41 1.0 th W10, S24 Sec. 2C] Interaction wtih Nuclei 329 TABLE 10 (Continued) Radiative Capture Cross Sections for Slow Neutrons Element At (barns) Neutron Energy § E T for Closest [| Observed Resonance (ev) References H 2oCa 0.41 P ~200,000 W10, H42, P23, C21 a 0.6 th W10, S24 48 1 P tt 2iSc 45 12 P W10, P23 (20 s) 10 P WW, G17 (85 d) 12 P,th W10, S24, ft 22T1 5.6 P W10, H42, C21, P23 46 0.57 P tt 47 1.62 P tt 48 7.98 P tt 49 1.80 P tt 50 0.14 P.th W10, S24, }t 23V 4.7 P WW, H42, C21, P23 51 4.5 th 2,700 W10, S24 24O 2.9 P 4,200 W10, R20, H42, C21, P23 50 16.3 P W10, tt 52 0.73 P W10, tt 53 17.5 P W10, tt 54 0.006 P,th WW, S24 2BMn 55 12.6 P 345 W10, H42, C21, P23 26^6 2.43 P ~10,000 W10, H42, C21, P23 54 2.1 P P22 56 3.1 P P22 57 0.5 P P22 58 0.36 th W10, S24 27CO 59 35 P 120 W10, H42, C21, P23 (10.7 m) 0.66 A W10, S24 (5.3 y) 22 th W10, S24 28 Ni 4.5 P 3,600 W10, H42, C21, P23 58 4.23 P P22 60 2.70 P P22 61 1.8 P W10, tt 62 15 P W10, tt 64 ~2 ih W10, S24 29C11 3.6 P ~500 W10, H42, C21, P23 63 4.29 P W10, tt 65 2.11 P W10, tt 30Z11 1.06 P 480 W10, H42, C21, P23 64 0.5 th W10, S24 68 (13.8h) 0.9 a W10, H72 (52 m; 0.1 th W10, H72 70 0.085 h W10, H72 330 The Neutron [Pt. VII TABLE 10 {Continued) Radiative Captube Cross Sections fob Slow Neutbons Element At (barns) Neutron Energy § E r for Closest || Observed Resonance Cev) References U siGa 82Ge S3A8 S4Se SsBr seKr 3?Eb ! 8 Sr 39Y 71 70 72 73 74 76 (59 s) (12 h) 75 74 76 77 80 (17 m) (59 m) 82 (67 s) (25 m) 79 (18 m) (4.4h) 81 78 80 82 83 84(4.4h) (10 y) 90* 2.9 1.4 3.4 2.4 3.3 0.94 13.7 0.60 0.35 0.03 0.2 4.2 12.0 44 82 40 0.5 0.03 0.05 0.004 6.5 8.5 2.9 2.3 28 0.3 95 45 205 0.1 0.06 0.06 <470 0.70 0.7 0.12 <200 1.2 1.3 0.005 <110 ~1 1.4 P th th P P P P P,th P P,th P,th P P P P th th th th P th th ih P th P P P th th th th P th th th P th th P P 100-500 W10, H42, P23 W10, H72 W10, S24 W10, H42, P23 W10, tt » U tt, S24 tt A33, tt S24, ft W10, H42, P23, C21, ft W10, H42, C21, P23 tt, S24 tt, A33 tt W10, S24 W10, S24 W10, A33 W10, A33 W10, H42, P23, C21 W10, H72, tt W10, H72, ft W10, S24 tt W10, tt tt tt tt W10, tt W10, tt W10, tt W10, tt W10, P23 W10, S24 W10, S24 W10 W10, H42, P23, C21, tt W10, S24 W10, S24 W10 W10 W10, P23 Sec. 2C] Interaction with Nuclei 331 TABLE 10 (Continued) Radiative Capture Cross Sections for Slow Neutrons Element At <K»,r) t (barns) Neutron Energy § E r for Closest I| Observed Resonance (ev) References H «Zr 0.20 P.th WW, R20, H42, P23, C21 90 ~0.1 P W10, tt 91 1.52 P WW, it 93 ~0.25 P W10, n 94 ~0.08 P WW, it 96 ~0.1 P WW, tt 4lNb 93 1.1 P WW, H42, C21, P23, tt 42M0 2.4 P WW, H42, P23, C21, ft 92 <0.001 th WW, S24 95 13.4 P WW, tt 96 1.2 P WW, it 97 2.1 P WW, tt 98 ~0.38 P.th WW, S24, tt 100 ~0.5 P.th WW, S24, tt 44R11 2.5 P 9.4 WW, H42, P23, ft 96 0.01 th W10, tt 102 1.2 th WW, S24 104 0.7 th WW, S24 «Rh 103 150 P 1.3 WW, H42, P23 (44 a) 137 th WW, S24 (4.3 m) 12 th WW, S24 46Pd 8 P WW, H42, P23, C21 108 11 th 24 WW, S24 110 0.4 th WW, S24 «Ag 60 P,th R20, H42, P23, tt 107 30 P 45 P22 109 84 P 5.1 P22 (22 8) ~100 th WW, S24, tt (225 d) 2.3 th WW, S24, tt 48Cd 2,400 »o R20, tt 3,500 P P23 106 1 th tt 110 0.2 th WW, G17 113 19,500 •0 0.18 WW, M39, D22, tt 114 (2.3 d) 1.1 th WW, S23 (43 d) 0.14 th WW, S23 116 1.4 th ~100 WW, S23 49ln 190 H,P R20, P23 113 (72 s) 2.0 th 3.8 WW, G17 (50 d) 56 th WW, S24 115 (13 s) 52 th 1.44 WW, S24 (54 m) 145 th WW, 824 332 The Neutron [Pt. VII TABLE 10 (.Continued) Radiative Capture Cboss Sections for Slow Neutrons "(n,y) t (barns) Neutron E r for Closest || Element a\ Energy § Observed References If Resonance (ev) soSn 0.65 P, »o H42, P23, C21, R20 112 1.1 A W10, S24 118 (279 d) 0.01 ft W10, M29, B55 120 0.22 ft W10, S24 122 (40 m) 0.30 ft W10, S24 124 (10 m) 0.6 ft W10, S24 (10 d) 0.15 ft W10, S24 siSb 6.4 P 5.8 WIO, H42, P23, C21 121 6.8 ft WIO, S24 123 (1.3 m) 0.03 ft WIO, D13, tt (21m) 0.03 ft tt (60 d) 2.5 ft W10, S24 62Te 4.5 P >300 W10, H42, P23, tt 120 68 P tt 122 2.7 P W10, H62, tt 123 390 P tt 124 6.5 P W10, H62, tt 125 1.5 P tt 126 (9. 3 h) 0.8 ft W10, S24 (90 d) 0.07 ft W10, S24 128 (72 m) 0.13 ft W10, S24 (32 d) 0.015 ft W10, S24 130 (25 m) 0.22 ft W10, S24 (30 h) <0.008 ft W10, S24 53I 127 6.7 P 20 WIO, H42, P23, C21 129* ~10 ft W10, tt 131* ~600 ft W10, tt 54X6 31 ft R20 128 <5 P tt 129 ~45 P tt 130 <5 P tt 131 120 P tt 132 0.2 ft W10, tt 134 0.2 ft W10, tt 135* 3.5 X 10 6 P U2 136 0.15 ft W10, tt b&Ca 133 29 P W10, H42, P23, R20, tt (3h) 0.016 th W10, S24 (2.3 y) 26 ft WIO, S24 135* ~15 ft W10, S57 137* <2 ft tt 6eBa 1.2 P WIO, H42, C21, P23 130 ~3 ft WIO, K3, tt 132 (>20 y) ~6 ft WIO, K3, tt 138 0.5 ft WIO, S24 139* 4 Ik W10, Yl, ft Sec. 2C] Interaction with Nuclei 333 TABLE 10 {Continued) Radiative Captuke Cross Sections for Slow Neutrons Element A\ <r(.n,y) t (barns) Neutron Energy § E r for Closest || Observed Resonance (ev) References U srLa 8.9 P W10, H42, P23 139 8.4 th WIO, S24 140 « ~3 th W10, K3 csCe 0.8 P WIO, H42, P23 138 ~0.4 th tt 140 0.27 th W10, K3, tt 142 0.85 th W10, K3, tt 69Pr 141 11.2 P (~10?) W10, P23 eoNd 44 P W10, H42, P23, tt 142 <12 P W10, H55 143 240 P W10, H55 144 <15 P W10, H55 145 <30 P W10. H55 146 1.8 th W10, B54, tt 148 3.7 th W10, B54, ft 150 <45 P W10, H55 eiPm 147* ~60 th W10, P4 B2Sm 6,500 to R20 10,000 P W10, H42, P23 149 ~50,000 P 0.096 W10, tt 151 ~7,000 P tt 152 150 th 10 W10, S24, B54, tt 154 5.5 th W10, S24 68Eu 4,200 P W10, P23 151 ~9,000 P -0.011 W10, H50, tt (9 h) 1,400 th W10, S24 152* 5,500 P W10, H50, tt 153 420 P 0.465 W10, H50, tt 154* 1,500 P WIO, H50, tt 155* 14,000 P W10, H50, tt 64Gd 36,000 P 0.03 WIO, H42, C21, P23 152 <125 a WIO, S24 155 70,000 p WIO, L4 157 120,000 p WIO, L4, tt 158 ~4 th WIO, B77, tt 180 ~0.15 th WIO, B77, tt « S Tb 159 44 P WIO, P23 eeDy 890 P -1.01;1.74 P23 1,150 »o WIO, B65 164 (1.3 m) 3,000 th WIO, tt (2.4h) 2,600 th WIO, S24, tt 165* 5,000 th WIO, K10 TABLE 10 (Continued) Radiative Captuee Ceoss Sections foe Slow Neutrons Element A\ <r(n,-y) t (barns) Neutron Energy § E r for Closest || Observed Resonance (ev) References IF !7Ho 165 64 P WlOy P23 esEr 166 P ~0.5 W10, P23 170 >7 th W10, B54 6»Tm 169 118 P W10, P23 voYb 36 P W10, P23 168 30,000 th W10, A35, tt 174 60 th W10, A35, tt 176 6.5 th W10, A35, tt 7lLu 108 P W10, P23 175 (3.7 h) 25 th W10, S24, tt 176 4,000 th W10, S24, ft 72Hf 120 P ~1.0 W10, H42, P23, C21 177 500 P tt 180 10 th W10, S24 7sTa 181 21 P 4.1 W10, H42, P23, C21 (16 m) 0.030 th W10, S24, tt (117 d) 21 th W10, S24 7»W 19 P W10, H42, P23, C21 180 ~2 th L19 182 4 183 7.4 184 2.1 th W10, S24 186 40 th 19.5 W10, S24, tt 187* ~80 th L19 76Re 84 P 2.3 W10, P23, tt 185 100 th tt 187 75 th tt 760s 14.7 P 6.5 W10, P23 184 ~20 th L19 190 8 th W10, S24 192 1.6 th W10, S24 193* ~190 th L19 77lr 440 P 0.64 W10, H42, P23 191 (1.5 m) 260 th W10, G15, tt (70 d) 740 th W10, S24, tt 193 130 th (1.3) W10, S24 7 8 Pt 8.1 P 11.5 W10, H42, P23, C21, tt 192 90 th W10, S24, tt 196 (18 h) 1.1 th W10, S24 (82 d) 0.055 th tt 198 3.9 th W10, S24 334 TABLE 10 (Continued) Radiative Capture Cross Sections for Slow Nbutbons Element At »(n,7) t (barns) Neutron Energy § E, for Closest |1 Observed Resonance (ev) References *[ 79A11 197 95 P.th 4.8 W10, H42, P23, S24, tt 198* ~16,000 th ft soHg 400 «o G20 340 P -2.0 W10, H42, P23, tt 196 3,100 P W10, 11 199 2,500 P W10, 11 200 <60 P W10, 11 201 <80 P W10, 11 202 3.0 th W10, S24, tt 204 0.43 th W10, S24, tt siTl 3.3 P 260 W10, H42, P23 203 8 th W10, S24, tt 205 0.10 th W10, S24, tt 82Pb 206 0.17 P 130,000 W10, H42, P23, C21, ft 208 0.0006 th 350,000 W10, tt 83B1 209 0.032 P W10, H42, tt (5d) 0.017 th W10, S24, tt joTh 232* 7.0 P.th R20, tt i 2 U 3.5 TO TJ2 235* 101 no D2 238* 2.80 m 11 U2 94PU 239 * 361 n V2 96Am 241* 890 P H29 (16 h) 570 P H29 t The atomic number A refers to the target nucleus. When more than one activity results from the radiative capture the half-life of the particular activity, to which the figures in that row pertain, is shown in parentheses. An asterisk indicates that the target nucleus is itself radioactive. % Unless otherwise indicated, the uncertainty can be assumed to be in the last significant figure. § The symbols have the following meanings: to = neutrons of velocity 2200 m/sec (E = -£§ ev) P = pile neutrons, usually indicating measurement by the "danger coefficient" method th = thermal neutron distribution for a temperature of ~300°K (usually indicates measurement by the "activation" technique) lj The closest observed resonance is not necessarily the closest resonance, since most elements have not been carefully investigated above ~I0 ev. Negative resonances are shown when known. H This compilation leans heavily on the excellent and complete collection of nuclear data by Way and co-workers (W10). The compilation of Ross and Story (R20), although unfortunately somewhat out of date, has been most useful. ft This table has been checked against a preliminary version of the extensive collection of the A.E.C. Cross Sections Committee, D. J. Hughes, Chairman. A number of values from that table have been added to this one. Furthermore, we have inclined somewhat to the choices, between alternative values, made by that committee. The responsibility for the choices, however, rests on our shoulders. We gratefully acknowledge our indebtedness to D. J. Hughes and his committee. Jt We are grateful to H. Pomerance of the Oak Ridge National Laboratory for making available unpublished measure ments of his group based on the "danger coefficient" method (P23) 335 336 The Neutron [Pt. VII (b) Intermediate Neutrons. For light and medium nuclei, (n,y) cross sections for intermediate neutrons are very small, since T y « r„. In heavy nuclei, the radiative capture cross sections are still quite appre- ciable in this energy range. The average (over many resonances) cross section is given by Eqs. (60), (62), (62a), and (63) : _ 2w^T n V y /600\ /iy\ 500\/iy II- — I barns (60a) (E is in ev). Thus, the cross section follows a 1/v law at low energies, and a 1/E law at high energies, the transition occurring when T n « T y ; for heavy nuclei, this occurs in the intermediate region. i 8 -I- -« 6 5 .(T t J -•- 4 -d 3 t »1.0 H St 8 ® ■" r -, b f 1 4 a ol on Reference No." d 3 Sj 111U Figure in Leger • 1 2 x l l © 4 o 5 8 6 5 4 3 0.1 ^7„(Mev) - 10 (1) R. Fields e( at., Phys. Rev., 71, 508 (1947). (2) H. Aoki, Proc. Phys.-Math. Soc. Japan, 21, 232 (1939). (3) J. R. Dunning et al., Phys. Rev., 48, 265 (1935). (4) E. Segre et erf. (unpublished). (5) J. Marshall and L. Szilard (unpublished). Fig. 21. Radiative capture cross section, a-(n,y), for iodine in the intermediate- neutron energy range. Also shown is the total cross section, 07. From Goldsmith, Ibser, and Feld (G20). (n,y) cross sections for intermediate neutrons have not been so exten- sively investigated as those for slow neutrons. Among the light and medium elements, fluorine, aluminum, and vanadium have been studied by Barschall and co-workers (see the compilation of Adair, A2); the <r(n,y) vs. E curves show the expected resonances; the magnitude of Sec. 2C] Interaction with Nuclei 337 <j(n,y) is small — of the order of millibarns. Curves of a{n,y) vs. E have been obtained, for a number of heavy elements, by Segre and co-workers; their results are included in the compilation of Goldsmith, Ibser, and Feld (G20). One of these, for iodine, is reproduced in Fig. 21. The cross section shows the expected behavior (see Eq. 60a). Measurements on fifteen isotopes, using Sb-Be neutrons (E ~ 35 kev), have been reported by Hummel and Hamermesh (H77). (c) Fast Neutrons. In the fast-neutron energy region, the behavior of <r(n,y) will differ from that described in the previous section for two main reasons: (1) With the inception of the possibility of inelastic scat- tering, the competition in the de-excitation of the compound nucleus becomes even less favorable to the radiative process. (2) For fast neu- trons the possibility of compound nucleus formation by neutrons of I > becomes appreciable, so that relationships such as Eq. (62b), based upon I = capture only, are no longer strictly valid. This factor tends to compensate for the decrease of radiative capture due to inelas- tic scattering, since the inelastic re-emission of I > neutrons is ener- getically unfavored. Hughes, Spatz, and Goldstein (H72) have made a systematic survey, covering 32 isotopes, of (n,y) cross sections at an energy of ~1 Mev. Their results are summarized in Fig. 22. The most significant features of these results are: (1) a rapid increase of c{n,y) with A (roughly expo- nential), from «1 millibarn at A « 35 to =200 mb at A « 110; (2) roughly constant (n,y) cross sections, of ~100 mb, for A > 120; (3) marked deviations from the norm for target nuclei containing neutron closed shells (e.g., Ba 138 , 82 neutrons; Pb 208 and Bi 209 , 126 neutrons). These magic number nuclei have anomalously small (n,y) cross sections of ~2-3 mb. Since the radiative capture cross sections at ~1 Mev are essentially inversely proportional to the compound nucleus level spacing at the excitation energy (e + 1) Mev (see Eqs. 60a and 37, with T ~ T n » T 7 ), the general dependence on atomic number is in reasonable agreement with expectation. The anomalous behavior of the magic number nuclei reflects their small binding energies for an additional neutron and, possibly, large level spacing. Hughes et al. (H72) have also surveyed the available data on the de- pendence of a(n,y) upon neutron energy, for energies between ~0.1 and ~10 Mev. While the results of the different investigators are in rather poor agreement with regard to the absolute values of a(n,y), the relative values seem to follow, roughly, a ~l/i? law. There are, however, a number of unexplained exceptions, notably In 115 , in which o(n,y) ap- pears to be constant over the energy range 0.1 to 1 Mev. 338 The Neutron [Pt. VII There are almost no data available on radiative capture in the very fast- and ultrafast-neutron regions. At these energies, a{n,y) is expected to be very small for all nuclei. 20 40 100 120 140 160 180 200 220 1.0 - 0.1 1 1 ' 1 i i 1 ■■T — 1 1 1 OLu o OAg n S & ™ O An - Kl. . -oi .. ° f /^Ag °Sb , H B / # • ^ J « Br / , ONci O / Nb / •* • V OK u ONd G^f.*^ * OBf • Gu/ Co°°/ 1 Mo 9Kr°M<, OPr /S Cu / °Ni OR u LaS^ k>Mn * .■°Ce OBi okA # / KrA OSr o OBa . /•$>v OFb Kb ~ /OA OXe . /ttci MgO/ PA\ i i i 1 1 1 1 , 1 , 40 60 100 120 140 160 A 180 200 220 Fig. 22. Activation (n,y) cross sections for fission neutrons (~1 Mev, average energy) vs. atomic weight, A. The points lying appreciably below the smooth curve all correspond to neutron numbers near or at one of the "magic" values — 50, 82, 126. We are indebted to Hughes, Garth, and Eggler (private communication) for this figure. Earlier results are reported and discussed by Hughes, Spatz, and Gold- stein (H72). 3. Charged Particle Reactions. After the capture of a neutron, the compound nucleus can sometimes decay by emission of a charged parti- cle. Among the possible reactions, (n,p) and (n,a) are most frequently encountered. The energy dependence of a charged particle reaction is governed by Eq. (42) : a(n,a) = <r c r o /r. (a) Slow Neutrons. For a charged particle reaction to take place with slow neutrons, it is necessary that the reaction be exoergic (Q > 0). Furthermore, if the reaction is to compete favorably, the available energy must be sufficiently great to allow appreciable penetration of the Coulomb barrier (the factor G a in Eqs. 32 and 37). These considera- tions limit the observable slow-neutron (n,p) and (n,a) reactions to light nuclei. The properties of the most important slow-neutron charged Sec. 2C] Interaction with Nuclei 339 particle reactions are summarized in Table 11. The cross sections in the third column refer to the isotopes involved in the reaction; these must be multiplied by the relative abundances (fifth column) to obtain the cross sections of the normal elements. TABLE 11 Properties of Exoergic (n,p) and (n,a) Reactions Reaction Q Value (Mev) Isotopic Cross Section at v = 2.2 X 10 B cm/sec (barns) References Relative Abundance of Isotope in Normal Element (%) He 3 (»,p)H 3 Li 6 (n,a)H 3 B w (n,a)Li 7 N 14 (n,p)C 14 Cl 35 (n,p)S 36 0.7637 4.785 2.791 0.626 0.62 5060 ± 200 910 ± 100 3770 ± 110 t 1.76 ±0.05 ~0.3 T16, C26, K14 T16, R20 T16, R20 T16, C26 W10 1 - 10 X 10" 5 7.4 18.83 100 75.4 f A more recent value of the B 10 (n,a) cross section is 3990 b (AEC Neutron Cross Section Advisory Group, AECU-2040, U. S. Department of Commerce, May 15, 1952). This cross section is of special significance since a majority of the quoted thermal neutron absorption cross sections, in this and in Table 10, are based on a comparison with boron absorption. Thus a change in the ac- cepted value of this cross section is directly reflected in a change, of equal frac- tional magnitude, in many of the other values quoted. The cross sections are given at a single neutron energy, 0.025 ev. Since these reactions fully satisfy the conditions for Eqs. (58) and (59), they can be assumed to follow the 1/v law in the slow-neutron range. Owing to the large level spacings of such light nuclei, the first resonances occur well into the intermediate- or even the fast-neutron region. Fur- thermore, owing to the large Q values (available charged particle energy) of these reactions, the reaction widths are essentially constant over a wide energy range. Thus, these reactions obey the 1/v law over a com- paratively broad energy region which, for the B 10 (n,a) reaction, for in- stance, extends to > 10 4 ev. (b) Intermediate Neutrons. For intermediate neutrons, charged parti- cle reaction cross sections depart from the 1/v law because (1) resonances are present in or close to the intermediate-energy region, and (2) the particle width, r , is no longer independent of the neutron energy. A resonance in the cross section for the formation of the compound nucleus is, of course, also a resonance in the reactions involved in the compound 340 The Neutron [Pt. VII nucleus decay, including the charged particle reactions (see, for in- stance, the compilation of Adair, A2, figures 8, 9, and 12). An interesting example of charged particle reaction resonances is N 14 , which has been investigated with good resolution for neutron energies between ~0.2 and 2 Mev, and is shown in Fig. 23. In addition to the (n,p) reaction, previously discussed, the N 14 (n,a)B u reaction is also in evidence. This reaction is slightly endoergic (Q= —0.26 Mev), and does not have an appreciable cross section below ~1 Mev. One of the most striking features of Fig. 23 is the apparent separation of the resonances (corresponding to the decay of the same compound nu- cleus) into predominantly (n,p) — e.g., 1.4 Mev— and (n,a) — 1.8 Mev — resonances. Although at first glance this may appear to be in contra- diction to the ideas of the compound nucleus picture — upon which we have leaned so heavily — the observations are, as will be seen from the following discussion, consistent with our present notions, if proper account is taken of the angular momentum and parity properties of the nuclear levels involved. 1 The nuclei involved are N 14 , C 14 , B 11 (all in their ground states), and N 15 (in various excited states). The spins of the nuclei are: N 14 , 1=1; C 14 , I = 0; B 11 , I = f; and their ground-state parities are: N 14 , prob- ably even (assumed 3 D, from the magnetic moment and to explain the long half-life for the C 14 beta-decay) ; C 14 , even; B 11 , probably odd (from the magnetic moment). Assuming these parity assignments, the parity of the level of N 1S , involved in the resonance, completely determines the I value of the captured neutron, even I values being associated with the states of even parity, and odd I values with odd states. From the laws of conservation of parity and angular momentum, the lowest possible I value of the emitted proton or alpha-particle is uniquely determined according to the following scheme involved) : V 1 3 7 \ 2 2 (J is the spin of the N 15 state 7 2 .. .. l p ■i- 1 • • * • la 1 1 1 3 .. l p 2 2 .. l a 2 2 2 4 l p 1 1 1 3 L 1 The author is indebted to Professor J. M. Blatt, who first called to his attention this possibility for explaining the N 14 resonance separation. A similar discussion has been presented by Johnson and Barschall (J5). Sec. 2C] Interaction with Nuclei 341 (siuBqiniui) o CO of 0) I N < CO fc • J2 "^ IS © O 4 CO t. a >• t8 3 » 00 fe* S "5 |2 IN s S3 <o .3 T3 §2 <N S © « ■3 &(? "*, H H §!3 ^^ * t. 9 5 73 CO CO 1 9"* eg o to H* fast n sectio tree, a IN vH ■§*£ ca >h 93 CO ediate total c ohnson a s 00 CO fc'g- d s for interm (A2). The ution, by J rH S3- tion dair reso 8 OS CO CO §^73 (N r-4 (n,a) re is from ery goo O CO sal 4) x-^ca £ PS ^"03 00 ! sT^a t3 O 53 C? 3 ^ ©& <o ■s 5s ea •2 § N ■* a' §1 o w 2 W ■St3 IB o3 CM ti ^ d o 1 - 1 »-> O . 1-9 M O si © [*( -o 342 The Neutron [Pt. VII It is evident that for low-neutron energies and, correspondingly, low energies of the emitted proton and (especially) alpha-particle (so that small I values are favored for both incoming and outgoing particle) the resonances divide into two groups: those for which J = \, favoring pro- ton emission; levels with J > § , which favor alpha-emission. The same sort of arguments can be carried through for different assumptions concerning the parities of the nuclei. In particular, the assumption of odd parity for the N 14 ground state (other states same as above) leads to a reversal in the division between proton-favored and alpha-favored levels; i.e., the absorption of a thermal neutron {l n = 0) is followed by the emission of an I = 1 proton, etc. Unfortunately, the available data do not permit a choice between the two possibilities for the parity of N 14 (J5, J7). (c) Fast Neutrons. As the energy available to the charged particle becomes greater, the Gamow barrier penetration factor approaches 1, and charged particle emission is less inhibited. Thus, reactions which, although exoergic or only slightly endoergic, have very small cross sec- tions for slow and intermediate neutrons become appreciable in the fast-neutron region. The N 14 (n,a) reaction, discussed above, is one such case. Another example is Ne 20 (w,a)O 17 (J6). There are a number of endoergic charged particle reactions whose thresholds, E t = — Q(A + 1)/A, fall in the fast-neutron energy range. The cross sections for these reactions have a characteristic energy de- pendence, rising rapidly (from o- = at the threshold) to a more-or-less constant value for energies greater than the "height" of the Gamow barrier. Figure 24 shows the measured (»,p) cross sections of two reac- tions (on P 31 and S 32 ) whose thresholds fall at ~1 Mev. The energy dependence of such "threshold reactions" is, at least for energies below the barrier heights, primarily determined by the probabil- ity for penetration of the Coulomb barrier by the emerging charged particle. The barrier penetration factor is a monotonically increasing function of the available energy (E — E t ) and, for a given value of E, strongly dependent on the angular momentum of the charged particle; the smaller the angular momentum, the greater the penetration factor. The energy dependence of the penetration factor has been discussed by a number of authors (B24, B34, B40), and by Morrison in Part VI. In general, if the reaction can proceed with the emission of charged parti- cles of zero angular momentum, it will prefer to do so, and the energy dependence near threshold will be determined by Go, the Gamow factor for I = particles. In this case the barrier height is zZe 2 Bo = — (80) Sec. 2C] Interaction with Nuclei 343 where z and Z are the atomic numbers of the outgoing particle and product nucleus, respectively, and the nuclear radius is given by (B34) : and R = 1.474* X 10~ 13 cm for protons R = (1.304* + 1.2) X 1(T 13 cm for alpha-particles (80a) Table 12 lists the properties of a number of useful (n,p) and (n,a) thresh- old reactions computed on the assumption of I = outgoing particles. TABLE 12 Properties or Fast-Neutron Threshold Reactions Based on computations by Feld, Scalettar, and Szilard, (F8), and Kiehn, (Kll) Product E (Mev) E (Mev) Reaction Nucleus E t (Mev) for for E t + £o Half-Life Go = 0.1 Go = 0.5 (Mev) P 31 (n,p)Si 31 2.7h 0.97 2.8 3.8 5.3 S 32 (n,p)P 32 14.3 d 0.96 3.0 4.1 5.6 AF(w,p)Mg 27 10 m 1.96 3.5 4.5 5.9 Si 28 (n,p)Al 28 2.3m 2.7 4.4 5.4 6.9 Fe 66 (w,p)Mn 56 2.6h 3.0 6.3 7.6 9.4 P 31 (»,a)Al 28 2.3m 0.91 6.6 8.3 9.8 AF(n,a)Na 24 14.9 h 2.44 7.5 9.1 10.9 The actual energy dependence of a given charged particle reaction is determined, in addition to the barrier penetration factor (in T a ), by the cross section for the formation of the compound nucleus, <r c . Among the important aspects of the dependence of <r c on E, the presence of resonances — especially in the light and medium nuclei, with which we are concerned — will be reflected in the reaction cross section; some of the resonances in a c may appear weakly or not at all in the reaction cross section because of the angular momentum and parity properties of the levels involved. Although the Gamow penetration factor for the outgoing charged particle becomes relatively constant (it slowly approaches one but is prevented from being strictly constant or equal to one by the increasing importance of higher angular momenta and their associated angular momentum barriers) after the available energy exceeds the barrier height, B , the reaction cross section will not remain constant as the neutron energy is indefinitely increased. In the very fast- and ultrafast- 344 The Neutron [Pt. VII 8 6 _' , P ff t _j J. 2 1 0.8 h ieference No. 0.6 l • 2 © 3 A 4 0.4 -*™i 7®^- -#<rp(s 2 ) 0.2 1 :: 0.1 B X 0.08 q ,_ 3 ^ A 0.06 T *£ [ 0.04 Hr- 1] 1 £ - C u 7 i -i Oi, X 4 b 0.01 L 0.008 0.006 I 5 * i r ~T [T - $- "T £ ■ j*\ A ^^ T T ~2 ' J 0.0 1 0.02 0.04 0.01 3 0.1 0. E I .30. tfev 4 0.6 > 1 2 3 4 6 8 10 (1) R. Fields ef a*., Phys. Rev., 71, 508 (1947). (2) H. Aoki, Proc. Phys.-Math. Soc. Japan, 21, 232 (1939). (3) E. D. Klema and A. O. Hanson, Phys. Rev., 73, 106 (1948). (4) R. F. Taschek (unpub- lished). (5) E. Bretscher et al. (unpublished). Fig. 24. The (n,p) cross sections of P 31 and S 32 in the fast-neutron region. Meas- urements between 1.4 and ~6 Mev, with poor energy resolution; these curves illus- trate the "threshold" behavior of endoergic charged particle reactions. Curves from Goldsmith, Ibser, and Feld (G20). 1 1 a (n,p) P SI (n,p)Si SI 1.5 > ^ a (n,p) 30-l(T 26 cm 2 T- S S2 (n,p)P s 3.5 MeV 3.0 3.5 MeV Fig. 24 (Continued). Good resolution measurements between ~2 and 3.5 Mev, showing the effect of resonances in the compound nucleus. Curve for P 31 due to Ricamo (R9); S 32 curve due to Liischer, Ricamo, Scherrer, and Zunti (L25). 345 346 The Neutron [Pt. VII neutron energy regions additional reactions, involving multiple neu- tron as well as charged particle emission, become energetically possible, and the cross section for any given charged particle reaction will, in the ensuing competition, decrease with increasing neutron energy. 4. Fission. For nuclei of A > 130 the binding energy per nucleon de- creases gradually with increasing A (Section 2B1). Consequently, most of the heaviest nuclei are energetically unstable against division into two fragments, i.e., M(A) > M(A — a) + M(a), provided that one of the fragments is a relatively tightly bound particle such as an alpha-particle or O 16 nucleus. However, as is well known, spontaneous nuclear disinte- grations are only (with two exceptions) observed in the heaviest, the naturally radioactive, elements. (Many nuclei, lighter than lead, which disintegrate by alpha-particle emission, have been produced artificially; but their lifetimes are much too short for them to be found in nature, even if they had been present in the original distribution of the elements.) It is, of course, the Gamow barrier against charged particle emission which impedes the spontaneous disintegration of the heavy elements, with the effect that alpha-emission is a relatively long-lived process and the emission of heavier fragments proceeds at an unobservably slow rate. There is, however, one mode of disintegration which involves the release of such a large amount of energy that the disintegration can, for the heaviest nuclei, proceed over the top of the barrier. This process is fission, or the division of the nucleus into two approximately equal fragments. Consider a heavy nucleus, say Z = 92, A = '240. The binding energy per nucleon is =7.5 Mev. Division into two fragments, e.g., Z t = Z 2 = 46 and A 1 = A 2 = 120, produces two nuclei which are approximately at the peak of the binding energy curve with, however, a considerable neutron excess, so that their average binding energy per nucleon is some- what less than that of the stable nuclei in this region, say = 8.3 Mev. The energy release is then ~240 X 0.8 ~ 200 Mev. The barrier height against separation of the fragments is ZiZ%e Z\Z 2 R^+R 2 = A x v * + A 2 A B = „ = ^ u ( 2mc2 ) ~ 20 ° M ev (80' ) [We have used the approximation R = (e 2 /2mc 2 )A H .] However, be- cause of the large charge of the fragments, the barrier is very "wide," and the lifetime against fission is a very strong function of the differ- ence between the barrier height and the available energy, being ex- ceedingly long for barriers only a few (~5-10) Mev higher than the reaction energy, and exceedingly short for energies above the barrier. Sec. 2C] Interaction with Nuclei 347 The greater the atomic number of the nucleus, the smaller is the margin of barrier height over reaction energy. This increase of instability with atomic number places an upper limit on the possible Z of the heaviest nuclei which can be found among the naturally occurring elements; nuclei with Z > 100 would be unstable against spontaneous fission (B46). 1 For the heaviest of the naturally occurring nuclei, then, stability against fission depends on the slight deficiency in the available energy as compared to the barrier height. The addition of only a relatively small energy can serve to push the reaction over the barrier. The ab- sorption of a neutron is one means of supplying energy, since capture is accompanied by release of the neutron's binding energy. If the binding energy is sufficiently great, as in the addition of a neutron to some odd- neutron nuclei, fission can be induced by the capture of a thermal neu- tron. Among the nuclei which undergo fission after thermal neutron capture are IT 233 , U 235 , Pu 239 (S42), Am 241 (C35), and Am 242 (H30). Since the energy release is so great, the fission widths, T f , for such nuclei should be essentially independent of the neutron energy. Thus, in the absence of close resonances, the thermal neutron fission cross section should follow a 1/v law. However, for such heavy nuclei the compound nucleus level spacing is expected to be relatively small. For U 235 , a t >,(n ,fiss.) = 545 barns (U2); additional data are given in Sec- tion 4D2. In most of the heaviest nuclei, however, the energy release accompany- ing neutron capture is insufficient to cause fission. For such nuclei, fission can be induced only if the neutron also carries with it a certain amount of kinetic energy, so that E + e + Q > B . The fission cross sections of these nuclei exhibit comparatively sharp thresholds. Many of the thresholds are in the fast-neutron region, and the corresponding nuclei make excellent "threshold detectors." Some of these nuclei, and their fission thresholds, are collected; in Table 13. Curves of o- fes . vs. E for U 238 and Np 237 are shown in Fig. 25. TABLE 13 Approximate Values of Fast Fission Thresholds Threshold Threshold Nucleus (Mev) Nucleus (Mev) Bi 209 60 TJ238 1.1 Th 232 1.3 Np 237 0.4 Pa 232 0.5 1 The experimental data on spontaneous fission are summarized by Segre [Phys. Rev., 86, 21 (1952)]. 348 The Neutron [Pt. VII 1.5 Np 23 7 • " m • V *f 0.5 0.5 1.0 1.5 E (Mev) (a) 2.0 0.7 0.6 SO-6 8 0.3 tj2S8 0.2 0.3 0.4 0.6 0.8 1.0 2.0 Neutron energy (Mev) (b) 3.0 4.0 6.0 F'g. 25. Fast fission cross sections vs. neutron energy, illustrating the "threshold" nature of the fission reaction, (a) Np 237 . From E. D. Klema, Phys. Rev., 72, 8S (1947). (b) Natural uranium; threshold due to U 238 fission (U2). Sec. 2C] Interaction with Nuclei 349 In the very fast- and ultrafast-neutron regions, fission can be induced in nuclei of smaller atomic number. Thus, nuclei down to platinum have been shown to undergo fission on bombardment by 84-Mev neu- trons (K7). The fission process has a number of unique features. In addition to the relatively tremendous amount of energy released per fission (Q ~ 200 Mev), the process is also accompanied by the emission of neutrons since the fragments, which are comparatively neutron-rich, are emitted with sufficient excitation energy to evaporate one or more neutrons. It is this feature which has made possible the achievement of a nuclear "chain reaction." Thermal neutron fission is actually asymmetrical, i.e., the two frag- ments have a tendency to be unequal in charge and weight, with the peaks in the fragment distribution occurring in the regions of A ~ 100 and A ~ 140. This tendency toward asymmetry is probably associated with the greater stability of nuclei containing the neutron magic num- bers 50 and 82 (G7). For fission induced by ultrafast neutrons, on the other hand, the fragment distribution appears to be more nearly sym- metrical. The change from asymmetry to symmetry with increasing bombarding energy is not inconsistent with an explanation in terms of the stability of magic number nuclei: a very highly excited nucleus will, before undergoing fission, evaporate a considerable number of neutrons; it will, therefore, not contain the ~132 neutrons necessary to produce nuclei close to two different magic numbers but will, rather, tend to split in such a way as to leave both fragments with as close as possible to 50 neutrons, i.e., symmetrically. 5. The (n,2ri) Reaction. The preceding discussion has covered all the exoergic reactions, and a few of the endoergic possibilities (inelastic scattering, charged particle emission), which can be induced by slow and intermediate neutrons. As the neutron energy is increased, through the fast, very fast, and ultrafast regions, a number of other threshold reactions become possible. One of the most important of these is the (n,2n) reaction, whose Q value is the binding energy of a neutron in the target nucleus. The product nucleus is an isotope of the target; in many cases it is radioactive, frequently decaying by positron emission. Neutron binding energies vary throughout the periodic table from 1.67 Mev in Be 9 to ~20 Mev in C 12 . Table 14 is a compilation of (n,2ri) thresholds. Most of the thresholds have been obtained from ob- servations on (y,n) reactions which, starting from the same target and leading to the same product nucleus, have the same Q values as the corresponding (n,2n) reactions. 350 The Neutron [Pt. VII TABLE 14 Thresholds fob (y,ri) and (n,2ri) Reactions; Neutron Binding Energies Target Element A (7,™) Threshold t (Mev) (n,2re) Threshold t (Mev) Product Nucleus Half-Life t References iH 2 2.226 ±0.003 3.34 stable M34 3 6.25 ±0.01 8.33 stable K16 3L1 6 5.35 ±0.20 6.2 S26 7 7.15 ±0.07 § 8.2 stable S26 4 Be 9 1.666 ±0.002 1.85 5 X 10 -14 „ s a M34 6 B 10 8.55 ±0.25 9.4 S26 11 11.50 ±0.25 12.6 stable S26 eC 12 ' 18.7 ±0.1 § 20.3 21m 3+ B6, M13 ?N 14 10.65 ±0.2 10.54 ±0.1 § 11.3 10 m P + M13 01 sO 16 16.3 ±0.4 17.3 2m f> + B6 9 F 19 10.40 ±0.3 § 10.9 1.9h H + 01 uNa 23 12.05 ±0.2 12.6 2.6y + S26 liMg 24 16.4 ±0.2 17.1 12 s 3 + M13, S26 25 7.25 ±0.2 7.5 stable S26 26 11.15 ±0.2 11.6 stable S26 13AI 27 12.75 ±0.2 13.2 7s fi + S26, M13 i<Si 28 16.8 ±0.4 17.4 5s P + M13 29 8.45 ±0.2 8.7 stable S26 15P 31 12.20 ±0.2 12.6 25 m /?+ M13, S26 ibS 32 14.8 ±0.4 15.3 3.2s + M13 34 10.85 ±0.2 11.2 stable S26 17CI ? 9.95 ±0.2 10.2 S26 19K 39 13.2 ±0.2 13\5 7.7 m 3 + M13 2oCa 40 15.9 ±0.4 16.3 Is M13 22T1 46 13.3 ±0.2 13.6 3.1h f) + 02 48? 11.6 ±0.3 11.8 stable S26 49 8.7 ±0.3 8.9 stable S26 23V 51 11.15 ±0.2 11.4 stable S26 uCv 50 13.4 ±0.2 13.7 42 m B + 02 52 11.80 ±0.25 12.0 25 d K S26 53 7.75 ±0.2 7.9 stable 1 S26 Sec. 2C] Interaction with Nuclei 351 TABLE 14 (Continued) Thresholds for (y,n) and (n,2n) Reactions; Neutron Binding Energies Target Element A Threshold t (Mev) (n,2n) Threshold t (Mev) Product Nucleus Half-Life % References 25MH 55 10.1 ±0.2 10.3 310 d K H35, S26 26Fe 34 13.8 ±0.2 14.1 8.9 m /3 + M13 56 11.15 ±0.25 11.3 2.9y S26 57 7.75 ±0.2 7.9 stable S26 27C0 59 10.25 ±0.2 10.5 72 d, 9.2 h K, + , IT S26 2sNi 58 11.7 ±0.2 11.9 36 h 0+ 01 61? 7.5 ±0.3 7.6 stable S26 29CU 63 10.9 ±0.2 11.1 10 m /3 + M13, S26 11.2 11.4 ±0.3 F40 65 10.0 ±0.2 10.2 12.8 h + , /3~, K M13, S26 3oZn 64 11.7 ±0.2 11.9 30 m + H35, S26 66 11.15 ±0.2 11.3 250 d K, j3 + S26 67 7.00 ±0.2 7.1 stable S26 68 10.15 ±0.2 10.3 stable S26 70 9.20 ±0.2 9.3 52 m p~ H35 3iGa 69 10.10 ±0.2 10.2 68 m (S + , K S26 71 9.05 ±0.2 9.2 S26 33AS 75 10.2 ±0.2 10.4 17 d + , /3~, K 01, S26 34Se 82 9.8 ±0.5 9.9 18 m 0~ B6 >9.8 57 m IT B6 ? 7.30 ±0.2 7.4 S26 ? 9.35 ±0.2 9.5 S26 3sBr 79 10.65 ±0.2 10.9 6.3 m 3 + M13, S26 81 10.1 ±0.2 10.3 18.5 m pT, P + M13, S26 3sSr 86 9.50 ±0.2 9.6 65 d K S26 87 8.40 ±0.2 8.5 stable S26 88 11.15 ±0.2 11.2 stable S26 4<)Zr 90 12.48 ±0.15 12.6 4.5m K01IT H35 12.0 ±0.2 12.1 78 h ,8+ 01 91 7.2 ±0.4 7.3 stable H35 4iNb 93 8.70 ±0.2 8.8 10 d /3 - S26 42M0 92 13.28 ±0.15 13.4 16 m, 17 s (3 + H35 97 7.1 ±0.3 7.2 stable H35 ? 6.75 ±0.25 6.8 S26 ? 7.95 ±0.25 8.1 S26 44RU ? 7.05 ±0.2 7.1 S26 ? 9.50 ±0.2 9.6 S26 352 The Neutron [Pt. VII TABLE 14 (Continued) Thresholds for (y,n) and (w,2n) Reactions; Neutron Binding Energies Target Element A (y,n) Threshold t (Mev)' (n,2n) Threshold t (Mev) Product Nucleus Half-Life J References 4sRh 103 9.35 ±0.2 9.4 210 d pr, 0+ S26 4 6 Pd ? 7.05 ±0.2 7.1 S26 ? 9.35 ±0.2 9.4 S26 47Ag 107 >9.5 24.5 m 0+ B6 109 9.05 ±0.2 9.1 2.3 m /3 - S26, B6 4«Cd 113 6.5 ±0.15 6.6 stable H35, S26 4»In 115 9.05 ±0.2 9.1 50 d IT; 72 a /3 _ , K, 0+ S26 5oSn 118 9.10 ±0.2 9.2 stable S26 119 6.55 ±0.15 6.6 stable H35. S26 124 8.50 ±0.15 8.6 40 m /3~ H35 5lSb 121 9.25 ±0.2 9.3 17 m /3 + M13 ? 8.95 ±0.25 9.0 S26 123 ~9.3 ~9.4 2.8d /3~ J4 52 Te ? 6.50 ±0.2 6.6 S26 ? 8.55 ±0.2 8.6 S26 5 si 127 9.3 ±0.15 9.4 13 d |3 - Ol, M13, S26 55Cs 133 9.05 ±0.2 9.1 7. Id K S26 56Ba J 6.80 ±0.2 6.8 S26 ? 8.55 ±0.25 8.6 S26 57La 139 8.80 ±0.2 8.9 stable; 18 h K, IT S26 ssCe 140 9.05 ±0.2 9.1 140 d K S26 142 7.15 ±0.2 7.2 33 d /3~ S26 69 Pr 141 9.40 ±0.10 9.5 3.5 m /3 + H35 eoNd 150 7.40 ±0.2 7.4 2h /3~ H35 7 3 Ta 181 7.6 ±0.2 7.6 8.2 h (T,K M13, S26, J4 74W ? 6.25 ±0.3 6.3 S26 ? 7.15 ±0.3 7.2 S26 75Re 187 7.3 ±0.3 7.3 93 h 0- K S26 77lr 193 7.80 ±0.2 7.8 70 d 0- S26 7 S Pt 194 9.50 ±0.2 9.5 4d K S26 195 6.1 ±0.1 6.1 stable P5, S26 196 8.20 ±0.2 8.2 stable S26 79AU 197 8.05 ±0.10 8.1 5.6d pr P5, H35, S26 Sec. 2C] Interaction with Nuclei 353 TABLE 14 (Continued) Thresholds for (y,n) and (n,2n) Reactions; Neutron Binding Energies Target Element A (t,«) Threshold t (Mev) (n,2n) Threshold t (Mev) Product Nucleus Half-Life J References soHg 201 6.25 ±0.2 6.6 ±0.2 6.3 6.6 stable H35 P5 siTl 203 205 8.80 ±0.2 7.5 ±0.15 8.8 7.5 12 d K 3y 0- S26 H35, S26, P5 82Pb 206 207 208 8.25 ±0.10 6.88 ±0.10 7.40 ±0.10 8.3 6.9 7.4 stable stable PI, P6 PI, S26, P5 PI, S26 83B1 209 7.4 ±0.1 7.4 M13, P5, S26 9oTh 232 6.35 ±0.04 6.4 26 h p~ H76, P5 92U 238 5.97 ±0.10 6.0 6.8 d 0- H76, P5 t The (n,2n) thresholds are computed from the measured (y,n) thresholds (or vice versa) according to the relationship Et(n,2n) = [(A + l)/AlEi(.y,n). t The half -lives and radioactivities are from the compilation of Way, Fano, Scott, and Thew (W10). § Thresholds computed from nuclear mass data and used to calibrate the gamma-ray energy scale. The most recent data on the thresholds for light elements are summarized by F. Ajzenberg and T. Lauritsen, Revs. Modern Phys., 24, 321 (1952). In light nuclei, the (n,2n) reaction competes with charged particle emission as well as with neutron scattering. In heavy nuclei, charged particle reactions are so strongly inhibited by the Coulomb barrier that the only important competing reaction — at least, for energies not too far above the (n,2n) threshold—is scattering, mainly inelastic at these energies. Assuming that this situation prevails, the value of the cross section, <x{n,2n), can be computed, given the energy distribution of the inelastically scattered neutrons, do(z,E), T(n,2n) e = E-E t d*(z,E) (81) That is, if the first neutron is emitted with sufficiently low energy (s < E — E t ), the residual nucleus will still have enough excitation to emit a second neutron and, since neutron emission is the most probable mode of decay for heavy nuclei at high excitation energies, it will almost always do so. At the excitation energies involved, and especially for the low-energy part of the spectrum, the distribution of inelastically scattered neutrons 354 The Neutron [Pt. VII is given, to a good approximation, by the Maxwellian distribution (Eq. 69). Thus, f-E-E, / s \ a(n,2n) = a c J \--j e -*T de (82) = <r c [l - (1 + y)e~»] where y = (E — E t )/T. For energies close to the threshold, i.e., y « 1, r(n,2ri) <fcy" 2T 2 {E - E t f (82a) The (E — E t ) 2 dependence of the (n,2n) cross section near threshold has been verified by Fowler and Slye (F40) for the Cu 63 (n,2n) reaction. 1 -1 .35 1 " .30 ~ .25 .20 (M - .15 s fc5" - .10 .05 11.0 12.0 13.0 14.0 ll.U iz.u la.u 14.U Neutron energy (Mey) (Uncertainty of energy scale ± 0.2_Mev) Fig. 26. Cross section for the reaction Cu^n^n^Ou 62 in the vicinity of the threshold, due to Fowler and Slye (F40). The left-hand scale gives the ratio of the Cu 63 (n,2n) cross section to the Cu 66 (n,y) thermal neutron cross section. The absolute cross section scale, on the right, is based on (0.56 ± 20%) barns for the thermal neutron cross section of Cu 66 in normal copper; it should be increased by ~16 percent (pri- vate communication from Fowler). Their cross-section measurements for neutron energies between E = E t = (11.4 =fc 0.3) Mev and <~14 Mev are shown in Fig. 26. This reaction is frequently used as a threshold detector of neutrons of energy E > 11.5 Mev. The C 12 (w,2n) reaction has also been used as a threshold detector for very fast neutrons. The cross section from threshold E t = 21 Mev to ~25 Mev increases rapidly with neutron energy in roughly the expected manner (S27). The expression for <x(n,2n), Eq. (82), has been derived on the assump- tion that the statistical theory, which leads to a Maxwell distribution of Sec. 2C] Interaction with Nuclei 355 the emerging neutrons, is valid in the region of excitation under con- sideration. The cross sections near threshold bear out this assumption. The statistical theory has been verified over a much wider range of excitation energies by the measurements of Bradt and Tendam (B58) on the relative cross sections for the (a,n) and (a,2n) reactions on silver and rhodium with 15-20 Mev alpha-particles, and by the observations of Temmer (T7) and Kelly and Segre (K8) on the excitation functions for the (a,ri), (a,2n), and (a,3n) reactions on indium and bismuth with alpha-particles up to 40 Mev energy. However, application of the statistical theory to (n,2n) reactions induced by very fast (and, especially, ultrafast) neutrons is subject to the limitation, pointed out by Weisskopf (W17), that neutrons of such high energy have a large probability of penetrating through the nuclear surface and hence may emerge from the nucleus before they have fully shared their energy with the rest of the nuclear constituents. The energy distribution of the emerging (first) neutron will, under these circum- stances, contain many more fast neutrons than predicted by the evapo- ration theory. This situation does not obtain for alpha-particle bom- bardment, since the Coulomb barrier impedes the emergence of the alpha-particle and helps to achieve the sharing of energy in a true compound nucleus. In any event, if, after the emission of the first neutron, the residual (target) nucleus is still sufficiently excited to emit a neutron, it will do so. The energy distribution of the second neutrons is determined by the level spacing of the product (final) nucleus at the excitation energy re- maining after emission of the second neutron. For incident neutron energies not very far above the (n,2ri) threshold, the energy available to the second neutron cannot be very large, and the spectrum of second neutrons cannot be assumed to follow the Maxwell distribution. In essence, the same considerations, which we have applied in the discus- sion of the energy distribution of inelastically scattered neutrons, will apply in this case. 6. Multiple-Particle Emission ; Spallation Reactions and Stars. The interaction of very fast and ultrafast neutrons with nuclei can be con- sidered to take place in two relatively distinct stages: (1) The initial interaction of the incident neutron with one or several of the nuclear constituents. In this stage the nucleons can be treated as relatively independent particles (S21), and the recoiling nucleons (as well as the incident neutron) have a non-negligible probability of leav- ing the nucleus, carrying off a relatively large fraction of the incident energy. 356 The Neutron [Pt. VII (2) Those nucleons which do not directly escape from the nucleus will rapidly share their energy among the nuclear constituents. In gen- eral, the resulting excited nucleus will have sufficient energy to evapo- rate more than one particle. The theory of the interaction of neutrons of E ~ 100 Mev with heavy nuclei has been developed by Goldberger (Gil). He has computed the energy and angular distributions of the emerging nucleons in the first stage of the interaction, as well as the distribution of excitation energies in the residual nuclei; his computations take into account the velocity distribution of the nucleons in the target nucleus (which he treats as a Fermi gas confined to the nuclear volume) . The process of ejection of nuclear constituents through direct inter- action with the incident particle is sometimes referred to as spallation. 1 The second or evaporation part of the reaction is usually referred to as star production, from the characteristic records that such reactions leave in nuclear emulsions which are sensitive only to relatively low-energy charged particles. One method of studying high-energy nuclear interactions is through the yields of the various products, usually radioactive (K19). How- ever, aside from the obvious limitation of this method (that only radio- active products can be detected), it has the disadvantage that the two stages of the reaction cannot be separated. A more satisfactory method for studying such reactions is by observ- ing the emerging particles, either in a cloud chamber or in a sensitive nuclear emulsion. In such investigations it is possible to separate the two stages of the reaction by the difference in the energy and angular distributions of the resulting charged particles. The products of the first stage have high energy and forward collimation (Gil). Those of the second stage have relatively low energy and are distributed uniformly with respect to the direction of the incident particle (L12). The appli- cability of the Goldberger model to high-energy nuclear reactions has been strikingly verified by the work of Bernardini, Booth, and Linden- baum (B19). An interesting result of such investigations is the observation of a considerable number of high-energy deuterons, emitted in the forward direction (H2, B74, Y3, M37). These appear to result from a "pick-up" process in which the incident neutron is joined by a nuclear proton, which happens to be moving in the same direction and with the proper velocity, to form a deuteron (C13, H51, C34). 1 Spall : A chip or fragment, esp. of stone. Webster's Collegiate Dictionary, fifth edition, G. and C. Merriam Co., 1948. Sec. 3A] Sources and Detectors 357 The interaction of ultrafast neutrons with nuclei has been studied in cosmic-ray investigations. Neutrons with energies exceeding 10 9 ev are present in the cosmic radiation. Many of these investigations have employed nuclear emulsions in which cosmic-ray stars are a common phenomenon. Le Couteur (LI 2) has shown that the energy distribution of the evaporation products in stars can be understood in terms of the statistical model; the energy of the emitted protons and alpha-particles is given by a modified Maxwell distribution, the modification arising from the inhibition of emission of low-energy charged particles by the Gamow barrier. Owing to the decrease with energy of the primary (n,p) and (n,n) cross sections, nuclei are relatively transparent to ultrafast neutrons of energy < 300 Mev. However, at neutron energies greater than the threshold for meson production (~285 Mev for neutrons on protons or neutrons at rest) the reaction cross section is expected to rise again. The available evidence indicates that nuclei are essentially opaque (re- action cross section « wR 2 ) for neutrons of energy > 10 9 ev. The products of the interaction of such neutrons with nuclei are, in the first stage, a number (increasing with neutron energy) of mesons (probably mostly pions) traveling in the forward direction, and fast nucleons (E ~ 100 — 1000 Mev) also in the forward direction; the residual nu- cleus, in the second stage, usually has sufficient energy to evaporate a large fraction of its particles (F10). SECTION 3. SOURCES AND DETECTORS: NEUTRON SPECTROSCOPY A. Introduction To investigate experimentally the properties of the neutron and its various interactions, it is necessary to have a neutron source and a neu- tron detector. If, in addition, it is desired to confine the investigation to neutrons of a specific energy, then either the source must emit mono- energetic neutrons or the detector must be sensitive only to neutrons of a single energy. The problems of neutron spectroscopy, and the dis- cussions of this section, involve the choice of the proper sources or de- tectors or both in the various neutron energy ranges. The resolution of a given experimental arrangement is a measure of the energy spread of the neutrons selected by the source and detector. As in other fields, good energy resolution must frequently be paid for by decreased intensity, if it can be achieved at all. On the other hand, poor resolution is by no means always an unmitigated ill. For example, for a comparison of intermediate- and fast-neutron cross sections with 358 The Neutron [Pt. VII the predictions of the continuum theory of Feshbach and Weisskopf, it is necessary to average over many levels of the compound nucleus; such an average is most easily and accurately achieved by using a source whose energy spread is sufficient to cover many levels of the compound nucleus. The term "resolution" is usually employed in a descriptive, qualita- tive connotation, mostly in conjunction with an adjective, such as "good" or "poor." The quantitative measure of energy resolution is the resolution function, R(E) = S(E)D(E), the product of the source strength and the detector efficiency. For many measurements, especially cross- section determinations in which there is considerable variation with energy (e.g., resonances), a knowledge of R(E) is indispensable for the interpretation of the experimental results. A considerable fraction of the literature on neutron cross sections is essentially uninterpretable, not so much because the experiments were performed with poor resolu- tion as because they were performed with sources of unknown energy distribution or detectors with unknown efficiency curves. However, although this criticism is sometimes applicable to recent work, it is certainly not fair to apply it to most of the early investiga- tions. To the pioneers in neutron physics, in the pre-World War II era, the number of sources and detectors available was severely limited. Although many of their cross-section measurements may not have been good in the present-day sense, or even quantitatively interpretable, their exploratory investigations were invaluable, for only through the accumulated knowledge of such explorations were present-day tech- niques made possible. In many instances what the early masters lacked in technique was more than compensated by the ingenuity of their ex- periments and the penetration and insight of their interpretation. The first available neutron sources were, naturally, those involved in the discovery of the neutron: (a,n) reactions on many elements, espe- cially beryllium, using naturally radioactive alpha-emitters. Some of this early work has been described in the introductions to Sections 1 and 2. The discovery of the photodisintegration of the deuteron by Chad- wick and Goldhaber (C8), and of beryllium by Szilard and Chalmers (S63) uncovered another source of neutrons, the (y,ri) reaction on these elements by gamma-rays from various natural radioactivities. These sources differ from the (a,ri) sources in that they provide lower-energy neutrons, usually in the intermediate-energy range, since (y,ri) reac- tions are endoergic. On the other hand, such sources are usually con- siderably weaker than (a,n) sources. However, with the production of strong artificially radioactive sources, at first through the use of parti- cle accelerators and more recently by neutron irradiation in nuclear re- Sec. 3A] Sources and Detectors 359 actors, a large variety of gamma-ray sources have become available for photoneutron production. Furthermore, the development of high- energy electron accelerators for the production of intense x-ray beams has added another means of intense photoneutron production, although these sources are not monoergetic like those produced by a nuclear gamma-ray. The first extensive investigation of photoneutron production in elements other than beryllium and deuterium was made by Bothe and Gentner (B52). They utilized the <~17 Mev gamma-rays produced in the Li 7 (p,y) reaction, using artificially accelerated protons of 0.4 Mev energy. Since then, a large number of investigations of (y,n) reactions have been carried out with these and other gamma-ray sources, and with x-ray beams from electron accelerators. The results of many of these are summarized in Table 14 (Section 2) and in the references therein indicated. Of all the nuclei investigated, only beryllium and deuterium have photoneutron thresholds of energy less than 6 Mev. The development of high-energy heavy-particle accelerators led to the discovery of a large number of new and useful neutron-producing reac- tions. In addition to providing monoenergetic alpha-particles of energy higher than those available from the natural radioactivities, with which it was possible to investigate further the (a,n) reactions on beryllium, boron, and other elements, it was found possible to utilize other acceler- ated nuclei, such as the proton and the deuteron, for neutron production. It was soon discovered that (d,ri) reactions have large yields at the deuteron energies available from accelerators. Crane, Lauritsen, and Soltan (C30) discovered the (d,n) reactions on lithium and beryllium. Soon afterward, Lawrence and Livingston (L8) extended the investiga- tion of (d,n) reactions to a number of other elements. At about the same time (all this in 1933 and 1934) Oliphant, Harteck and Rutherford (03) discovered the d + d reaction; this reaction has been of great importance, since it provides a strong source of monoenergetic neutrons with com- paratively low-energy deuterons, enabling the use of relatively low- voltage particle accelerators for neutron sources. Other (d,ri) reactions have comparable or higher yields at deuteron energies of a few Mev, but the resulting neutrons are heteroenergetic (with the exception of the d + t reaction). Another reaction type of great importance to neutron spectroscopy is the (p,n) reaction. Crane and Lauritsen (C31) discovered the ~Li(p,ri) reaction, which has since been extensively used for a monoenergetic neu- tron source. However, this reaction does not really yield monoenergetic neutrons, since it also gives rise to a second neutron group, due to an excited state of Be 7 , 435 kev above the ground state. Fortunately, the 360 The Neutron [Pt. VII second group has a comparatively low yield (<10 percent) for protons up to -~4 Mev. A number of other (p,ri) reactions have been studied and used from time to time for neutron sources, but none so extensively as the Li 7 (p,n) reaction. All the above-mentioned neutron sources yield energies in the inter- mediate, fast, and very fast ranges. Until the advent of ultrahigh- energy accelerators, the only source of neutrons in the ultrafast-energy range was from the cosmic radiation. The presence of high-energy neutrons as a component of cosmic rays was established by the experi- ments of Rumbaugh and Locher (R25) and of Fiinfer (F51), and has been used by many investigators to study the nuclear interactions in- duced by ultrafast neutrons. Although particle accelerators are at present capable of producing neutrons of energy up to ~400 Mev, and higher-energy accelerators (a few Bev) are now being constructed, cos- mic radiation still remains the only source in immediate prospect for neutrons of energies of, say, > 10 Bev. At the other extreme, the slow-neutron region has been most exten- sively investigated. Sources of slow neutrons almost invariably have their origin in fast neutrons, slowed down in paraffin or some other material containing light elements. Since such sources yield a broad distribution of neutron energies, it is necessary to employ some form of neutron monochromator for studies requiring monoenergetic neutrons. A number of monochromators have been, and are being, developed, and their effective range has been slowly pushed up toward the intermediate- energy region, so that there now remains only a small gap between the monoenergetic neutrons available from charged particle reactions and from slow-neutron monochromators. 1 The availability of very great neutron intensities from nuclear reactors has provided a great impetus to the development of more effective neutron monochromators of greater range and flexibility. The development of neutron detectors has rapidly followed the exten- sion of knowledge of neutron reactions; practically every new discovery has led to a new means of neutron detection. Thus, the observation of proton recoils by Curie and Joliot (C37), made even before the identifi- cation of the neutron as a new particle, led to the technique of observing neutrons in ionization chambers, electroscopes, and cloud chambers by lining these instruments with paraffin. The discovery of charged parti- cle reactions in lithium, boron, and nitrogen enabled the detection of 1 Actually, the gap is being closed from the intermediate-energy end as well. Thus, Hibdon, Langsdorf, and Holland [Phys. Rev., 85, 595 (1952)] have succeeded in studying the 2-25 kev range with an energy resolution of 2 kev, using the Li(p,w) reaction. Sec. 3B] Sources and Detectors 361 neutrons through the incorporation of these substances in ionization chambers, proportional counters, and cloud chambers. These and other reactions, as well as proton recoils, can also be observed in nuclear emulsions. The discovery of neutron-induced radioactivity provided still another means of neutron detection which could be used to investigate specific energies or energy ranges — thermal neutrons through 1/v capture cross sections, specific slow-neutron energies through various resonances, fast neutrons by means of threshold reactions. As neutron reactions have been further understood, and as various techniques of charged particle counting have been improved and extended, the variety of neu- tron detectors has increased until now it is possible to find a suitable neutron detector at almost any energy. This is not to say that detector problems are negligible in neutron studies; as is so often the case, ease of detection is in direct proportion to the available intensity, so that the source and detector problems of neutron spectroscopy go hand in hand. Especially in the fast-, very fast-, and ultrafast-neutron energy ranges, detectors are of relatively low efficiency, and available sources are never quite strong enough, so that neutron spectroscopy in these energy regions still presents difficult problems. Nevertheless, it seems fair to summarize by saying that the available techniques of neutron spectroscopy allow an almost complete coverage of the range of energies from to ~300 Mev with relatively few signifi- cant gaps, and that, with foreseeable extensions of available techniques, the existing gaps should soon be closed. B. Neutron Sources 1. Radioactive (a,ri) Sources. The discovery of the neutron involved the reaction 4 Be 9 + 2 He 4 -» 6 C 12 + on 1 induced by bombarding beryllium with a-particles emitted by the natu- rally radioactive elements. Although many other neutron-producing reactions have since been discovered, the above reaction is still the basis for some of the most extensively used neutron sources. According to available mass values (T16, B2), this reaction is exoergic, with a Q value of 5.65 Mev. Thus, starting with the polonium alpha- particles (energy 5.30 Mev), the emergent neutrons should have a spread of energies between 10.8 Mev (outgoing neutron in the same direction as the incoming alpha) and 6.7 Mev (outgoing neutron in the opposite direction from the incoming alpha). 362 The Neutron [Pt. VII However, the neutrons observed in the bombardment of beryllium with polonium alpha-particles have a considerably greater energy spread, ranging from the above maximum to energies well below 1 Mev. The observed energy spread arises from two causes: (1) In the above reaction, the C 12 nucleus may sometimes be left in an excited state, re- sulting in less available energy for the outgoing neutron. This possi- bility will, for monoenergetic incident alpha-particles, result in the appearance of groups in the spectrum of the outgoing neutrons, each group corresponding to an excited state of the C 12 nucleus. Evidence for the existence and energy values of the C 12 levels is summarized by Hornyak, Lauritsen, Morrison, and Fowler (H67). (2) In most (a,n) sources, the thickness of the beryllium target is large compared to the range of the impinging alpha-particles. (The range of a polonium alpha-particle is 3.66 cm in standard air.) Since the cross section for the neutron-producing reaction is small compared to the cross section for energy loss by collisions with atomic electrons, very few nuclear processes occur while the alpha-particle has its full, initial energy. Thus, even if all the reactions led to the ground state of C 12 , the out- going neutrons in the forward direction would have an energy spread ranging from the maximum (10.8 Mev) down to 5.2 Mev (correspond- ing to zero incident alpha-particle energy). The energy spectrum of Po-a-Be neutrons is further complicated as a result of the variation of the reaction cross section with the incident alpha-particle energy. Thus, the necessity for the alpha-particle to penetrate through the potential barrier of the beryllium nucleus de- creases the neutron yield for low-energy alpha-particles; the height of the potential barrier is ~3.7 Mev. Furthermore, the level structure of the compound nucleus, C 13 , leads to resonances in the cross section (H67). As a result of the effects discussed above, the neutron spectrum from a (thick target) Po-a-Be source is complex, and cannot be predicted in detail. A number of attempts have been made (A27) to measure the neutron spectrum from such a source using proton recoils, in nuclear emulsions, as a neutron detector. The result of a recent measurement, due to Whitmore and Baker (W20), is shown in Fig. 27. Among the alpha-emitting radioactive elements, polonium is compara- tively difficult to obtain in quantities sufficient to produce strong neu- tron sources. The elements radium and radon are, however, commer- cially available in sufficient quantity so that they are most frequently used for neutron sources. Radium has the advantage of a very long half- life (~1600 years, as compared to 3.825 days for radon and 138 days for polonium), which makes it particularly suitable for long-lived sources. Sec. 3B] Sources and Detectors 363 On the other hand, polonium and, even more so, radon require a con- siderably smaller mass of beryllium in the source mixture to approxi- mate a thick target (since the mass per unit radioactive strength of a radioactive element is proportional to its half-life). In particular, radon requires quite small quantities of beryllium for high neutron yields, and presents no difficulty in mixing, since it is a noble gas and diffuses uniformly through powdered beryllium; thus, Rn-a-Be sources \ V- T N 4 5 6 7 Neutron Energy (Mev) 10 11 Fig. 27. Energy distribution of neutrons from a Po-a-Be source (W20). can be made quite small. However, owing to the inconvenience of working with a gas of such short half-life, radon is now seldom used for neutron sources. Po-a-Be sources have the added advantage that in the decay of polonium there is a comparatively negligible gamma-ray emission, which makes the handling of such sources relatively simple. The decay products of radium, on the other hand, emit a prodigious quantity of gamma-radiation, and suitable precautions for protection against these radiations must be observed in the handling of these sources. Never- theless, owing to the conveniences of availability and long life, Ra-a-Be neutron sources are very widely used. Another disadvantage of polonium as compared to radium is that it is more difficult to manipulate, despite the absence of gamma-radiation. One method of preparation of a Po-a-Be source, described by Spinks and Graham (S49), consists in sandwiching a platinum foil, on which the polonium is deposited, inside a cylinder of beryllium. Such a source 364 The Neutron [Pt. VII has a smaller yield per curie (~3^) than an intimate mixture, and the neutron emission is not isotropic. During World War II considerable experience was obtained in the preparation of radium-beryllium mixtures. 1 The preparation and handling of such sources has been described by Anderson and Feld (A26) and, in greater detail, by Anderson (A27). Most of these con- sist of an intimate, physical mixture of radium bromide and beryllium metal powder, pressed into pellets of density ~1.75 g/cm 3 . In addition to their small size (thereby more closely approximating a point source), pressed sources are more likely to remain constant in time because of the greater physical stability of the mixture. The neutron spectrum from Ra-a-Be sources is even more complex than that of Po-a-Be, owing to the variety of alpha-particles emitted by radium and its decay products, as shown in Table 15. The presence of polonium alpha-particles in a Ra-a-Be source is governed by the decay of radium D, with a 22-year half-life. Hence, in relatively young (a few years old) sources, this last alpha-particle is not appreciably present. TABLE 15 Alpha-Particle Energies from Radium and Its Decay Products Alpha-Emitter Half-Life Energy (Mev) Ra 1620 y J 4.795 (93.5%) 14.611 (6.5%) Rn 3.825 d 5.486 RaA 3.05 m 5.998 RaC 1.5 X 10- 4 s 7.680 RaF(Po) 138.3 d 5.300 The rest of the alpha-particles are fully present after a few weeks. All the following discussion (spectrum, yield) is concerned with such young sources. Since both the spectrum and yield of Po-a-Be sources are comparatively well known, it is easy to take into account the changes in the source due to the accumulation of polonium. The Ra-a-Be neutron spectrum has not been nearly so extensively investigated as Po-a-Be, mainly because of the difficulties of neutron measurement in the accompanying high gamma-ray background. Its properties may be roughly summarized as follows (A27): The fast- neutron spectrum extends to a maximum energy of ~13 Mev, with a broad peak at ~4 Mev. There appears to be a substantial group of intermediate-energy neutrons, but there is considerable uncertainty as 1 A large number of these sources were prepared in the laboratory of the Radium Chemical Company, 570 Lexington Ave., New York City. Sec. 3B] Sources and Detectors 365 to their amount and energy. Various estimates of the yield of inter- mediate neutrons range from ~10-30 percent of the total yield. These have been ascribed to the reaction He 4 + Be 9 -* 3He 4 + n 1 (in which a number of intermediate steps have been omitted). The low-energy group may also arise, in some part, from a (y,n) reaction on beryllium. In any event, the low-energy group does not seem to be present in Po-a-Be sources, owing either to the absence of higher-energy alpha-particles (those from RaC are assumed mainly responsible for the 3a reaction) or of gamma-rays, or both. Other light elements beside beryllium can be used to produce neutrons through (a,n) reactions. Thus, a pressed Ra-a-B source has been pre- pared, and its spectrum and yield studied (A27). Both B 10 and B 11 undergo exoergic (a,n) reactions (Q = 1.18 and 0.28 Mev, respectively), the latter being responsible for most of the neutron yield. The spectrum is comparatively simpler than that of a Ra-a-Be source, rising rapidly to a maximum at ~3 Mev, and then falling rapidly to zero at ~6 Mev. There does not appear to be any appreciable intermediate-energy component. The reaction F 19 (a,n) is believed (from mass values) to be slightly exoergic, by <0.5 Mev. Bretscher, Cook, Martin, and Wilkinson (B62) have prepared a source composed of the relatively stable com- plex, RaBeF 4 , which they suggest for a standard neutron source, since the characteristics of the complex are not expected to change appre- ciably with time. The yields of the various sources discussed above have been studied by a number of methods (A27). In general, for a given mixture of TABLE 16 Yields or Radioactive (a,n) Sources fob Intimate Mixtures of an Alpha-Emitter and a Neutron-Producing Material Source Y (10 6 neutrons/curie • sec) Ra-a-Be 17 Rn-a-Be 15 Ra-a-B 6.8 Po-a-Be 3 RaBeF* 2.53 f t Since the mixture is fixed, the value given is that of an actual source com- posed of the complex. This yield could be improved by adding beryllium, but this would nullify the purpose of such a source, namely, the elimination of pos- sible changes due to alteration of the physical composition. 366 The Neutron [Pt. VII some alpha-emitting compound (a-mat) and neutron-producing material (X), the yield is given by the relationship Y=Y M(X) M(X) + M(a-mat) (83) Values of F (yield for an <x> ratio of X to a-mat) are given in Table 16. (<x,ri) yields from various materials have been extensively investi- gated, and are summarized by Anderson (A27). A number of these in- vestigations have employed thin sources and targets (S61, W3, H9), while others have measured thick target yields. Most of the investi- gators employed polonium alpha-particles, varying their energy by changing the pressure of gas between source and detector. The thin target (a,n) cross section for beryllium (due to Halpern, H9) is shown in Fig. 28a. Figure 28b shows the results of Segre and Wiegand on the thick target yields of beryllium, boron, and fluorine. Thick target yields, for (artificially accelerated) 9-Mev alpha-particles, have been measured by Ridenour and Henderson (Rll), and for 30-Mev alpha- TABLE 17 Neutron Yields fob Polonium Alpha-Particles on Thick Targets Yield per 10 6 Alphas Element Roberts Segre and Wiegand (A27) Walker Halpern Szalay (A27) (W3) (H9) (S61) Li 2.6 4.7 Be 80 73 50 B 24 19 19 C 0.11 N 0.01 0.07 F 12 10 Na 1.5 Mg 1.4 0.5 Al 0.74 0.25 0.22 Si 0.16 CI 0.11 A 0.38 Sec. 3B] Sources and Detectors 367 0.2 0.6 1.0 a range in air (cm) 1.4 1.8 2.2 2.6 3.0 3.4 3.8 0.44 0.40 0.36 "£•0.32 a a 0.28 x> "^0.24 o '•§ 0.20 o> So.16 CO J 0.121- 0.08 0.04 I I j 1 - Be ' 1 ■] ! -1 l ■ i -v — 1 i | ( I- " - • •M • • • • • • • • • • • • • - • • • • • • • • - • - ••)• i i 1 1 1 1 1 1 1 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Energy (Mev) (a) 4.5 5.0 80 70 Jso a, J 50 fl 1 Beryllium 2 Boron 3 Fluorine '2 3 2 3 Energy (Mev) (b) Fig. 28. (a,n) yields vs. neutron energy for a number of elements, (a) Thin target (a,n) cross section of beryllium, due to Halpern (H9). (b) Thick target (a,n) excita- tion functions of beryllium, boron, and fluorine, due to Segre and Wiegand (A27). 368 The Neutron [Pt. VII particles by Allen, Nechaj, Sun, and Jennings (A8), who also measured the angular distributions of the neutrons. The results of a number of investigations of thick target yields from various elements for polonium alpha-particles are summarized in Table 17, in which the data of Rob- erts and of Segre and Wiegand are taken from Anderson's summary (A27). The results of the last three investigators are obtained by inte- gration of their thin target yield curves. 2. Photoneutron Sources. Photons can cause neutron emission from nuclei, provided that their energy is greater than the neutron binding energy. In so far as the (y,n) reaction is concerned, the periodic table is conveniently divided into two groups: (1) deuterium and beryllium, and (2) all the rest. The former have (y,n) thresholds of (2.226 ± 0.003) and (1.666 ± 0.002) Mev, respectively. The latter have thresholds in excess of 6 Mev. Since no radioactive nuclei which emit gamma-rays of such high energy are known, radioactive (y,n) sources employ only beryllium or deuterium. The (prompt) gamma-rays from various nu- clear reactions or the x-rays obtainable from high-energy electron accelerators can be used to obtain neutrons from other nuclei as well. (a) Radioactive (y,n) Sources with Deuterium and Beryllium. The bombardment of deuterium or beryllium by monoenergetic gamma-rays results, for a given angle 6 between the emitted neutron and incident gamma-ray, in monoenergetic neutrons, according to the relation- ship (W8) B ' - (^T 2 ) ET 2 E y -Q- 5^E y 1862(A - 1) 2(A - l)(E y - Q) 931 A 3 + 5 cos 6 (84a) (84b) E n is the neutron energy; E y , the gamma-ray energy; Q, the neutron binding energy, all in Mev. A is the mass number of the target nucleus. In almost all practical photoneutron sources, for reasons of intensity, the gamma-ray source is surrounded by beryllium or deuterium, so that the neutrons have an inherent energy spread corresponding to an iso- tropic distribution in the angle 0, AE n = 25 (84c) For 100-kev neutrons, AE n /E n « 4 percent for beryllium, and « 25 percent for deuterium. The relative spread decreases with increasing neutron energy. The above is, however, not the main cause of energy spread. For most sources, a larger uncertainty in the neutron energy arises from the Sec. 3B] Sources and Detectors 369 fact that considerable quantities of beryllium or deuterium must be used to obtain usable neutron intensities. Since both beryllium and deuter- ium are quite light, and since the neutrons have a non-negligible proba- bility of undergoing a scattering before emerging from the source, an appreciable energy spread may thereby be introduced. In addition, neutron scattering in the source has the effect of reducing the mean energy of the emerging neutrons. Furthermore, the gamma-rays can lose energy by Compton scattering and then produce neutrons, an effect which introduces a further uncertainty in the neutron energy. Since the Compton cross section is ~10 3 times that for photodisinte- gration, a source containing an infinite quantity of beryllium or deuter- ium will produce only ~1 neutron per 1000 gamma-rays. TABLE 18 Photoneutbon Soukces Standard Yield | Source Tl A £ T (Mev) E n (Mev) (10 4 neutrons/ sec • curie) (1 gram at 1 em) Na 24 + Be 14. 8h 2.76 0.83 13 Na 24 + D 2 2.76 0.22 27 Mn 66 + Be 2.59h 1.81,2.13,2.7 0.15,0.30 2.9 Mn 66 + D 2 2.7 0.22 0.31 Ga 72 + Be 14. lh 1.87,2.21,2.51 (0.78) f 5 Ga 72 + D 2 2.51 0.13 6 Y 88 + Be 87 d 1.9,2.8 0.158 ± 0.005 § 10 Y 88 +D 2.8 (0.31) 0.3 In 116 + Be 54 m 1.8,2.1 0.30 0.82 Sb 124 + Be 60 d 1.7 0.024 ±0.003 § 19 La 140 + Be 40 d 2.50 0.62 0.3 La 140 + D 2 2.50 0.151 ± 0.008 § 0.8 MsTh + Be 6.7y 1.80, 2.62 0.827 ± 0.030 § 3.5 MsTh + D 2 2.62 (ThC") 0. 197 ± 0.010 § 9.5 Ra + Be 1620 y 1.69, 1.15,1.82, 2.09, 2.20, 0.4# a mess 3.0 Ra + D 2 2.42 0.12 0.1 t The standard yield is taken to be that of 1 gram of beryllium or heavy water at 1 cm from 1 curie of the substance indicated. % Values in parentheses are estimates. § Due to Hanson (H33). 370 The Neutron [Pt. VII These factors, as well as other aspects of the production and use of photoneutron sources, are discussed in considerable detail by Watten- berg (W8,W7). Table 18 summarizes the properties of available radio- active (y,n) sources. The characteristics of the cross section for the D(-y,n) reaction have been discussed in Section 1. Considerably less is known concerning the Be(y,n) cross section. The available evidence, both experimental (R26) Fig. 29. Design of the primary photoneutron source standard at the National Bureau of Standards (C42). The beryllium sphere is 4 cm in diameter and holds, at the center (C), a 1-gram capsule of radium. and theoretical (G32), indicates that the cross section passes through at least one maximum and one minimum as the gamma-ray energy is increased from threshold to 2.76 Mev. Prior to the extensive availability of strong artificially produced gamma-ray sources, Ra-y-Be sources were extensively used to provide intermediate-energy neutrons. Following a suggestion of Gamerts- f elder and Goldhaber (G2), such a source has been prepared by Curtiss and Carson (C42) at the National Bureau of Standards, to serve as a permanent neutron standard. Since a Ra-7-Be source does not require mixing of the radium and beryllium (with the attendant possibilities for physical change), its neutron output should not vary with time. In the standard source, a pressed radium bromide pellet is placed at the center of a carefully machined sphere of beryllium metal; the design of the standard source is shown in Fig. 29. The absolute yield of such a source could be obtained, without the necessity of any neutron measurement, by a method developed by Sec. 3B] Sources and Detectors 371 Paneth and Gluckauf (P2, G8). They measure the total accumulation of helium 1 after a known time of irradiation of the beryllium. An interesting application of the photodisintegration process is as a gamma-ray detector which is completely insensitive to gamma-rays of energy below the photodisintegration threshold (P8a). Myers and Wat- tenberg (M42) have used this device to detect the presence of a small component of "cross-over" gamma-rays when two or more gamma-rays, both of which are below the photodisintegration threshold of beryllium or deuterium, are emitted in cascade. (b) Photoneutrons from High-Energy Gamma-Rays and X-Rays. Pho- toneutron reactions with high-energy gamma-rays have been investi- gated in a large number of elements. Following the work of Bothe and Gentner (B52), the ~17-Mev Li(p,y) gamma-rays have been used to study the (y,ri) cross sections of many elements, by Waffler and Hirzel (Wl) and by McDaniel, Walker, and Stearns (M12). However, the most extensive investigations have employed x-ray beams from electron accelerators, mainly betatrons. The x-rays are produced by causing the accelerated electrons to strike a target, usually of some heavy element. The resulting radiation is allowed to fall on the material under investigation. (7,71) reactions are detected either through direct observation of the neutrons, or by de- tection of the radiations from the product (usually /3 + -radioactive) nuclei. The shape of the x-ray spectrum from an electron accelerator depends on the target thickness. For relatively thin targets, the distribution of x-ray quanta follows a bremsstrahlung spectrum, at least for energies not too far below the maximum (electron) energy, dN(E y ) = —1 (85) Ey The measurement of (y,n) cross sections with such x-ray beams involves the complication of dealing with a heteroenergetic source. However, if the electron energy can be varied, (y,n) yields can be measured as a func- tion of the maximum x-ray energy. The results of three such studies, due to Diven and Almy (D15), are shown in Fig. 30a. Such curves can be interpreted in terms of the (y,n) cross section vs. E y , provided that the x-ray spectrum is known and the x-ray intensity (the value of k in Eq. 85) is calibrated. 1 The reaction is Be 9 (7,n)Be 8 ; Be 8 -> 2He 4 . 372 The Neutron [Pt. VII Some curves of <r(y,ri) vs. E y are shown in Fig. 30b. The striking fea- ture of such cross-section curves, first noted by Baldwin and Klaiber (B7), is the strong resonance shape. This shape has been observed for all the nuclei studied, although the positions of the maxima and the 16 14 12 za a g 8 B a) Ag 1 07,109 Cu 63 /A J 27 10 12 14 16 18 20 Maximum x-ray energy (Mev) • 22 24 Fig. 30a. Relative neutron yields vs. maximum x-ray energy for three nuclei, due to Diven and Almy (D15). The ordinates are in arbitrary units. resonance widths vary from nucleus to nucleus (D15, M13). Particu- larly accurate work, determining the resonance constants for many nuclei, has been done by the Saskatchewan Group (J4, K4). An explanation of these resonances has been advanced by Goldhaber and Teller (G16). They have postulated the possibility of dipole vibra- tions in which the protons (as a whole) oscillate with respect to the neu- trons in the nucleus. The general features of the process of dipole radia- tion capture have been derived by Levinger and Bethe (LI 6). Sec. 3B] Sources and Detectors 373 Neutron yields, due to the (y,n) reaction, have been measured for many elements and at many x-ray energies, up to 330 Mev (P31, T8). Neutron sources of considerable strength can be obtained, through (y,n) reactions, from electron accelerators. Thus, by using the elec- trons from a 3.2-Mev linear accelerator to produce gamma-rays in a lead target, and by irradiating a heavy water (or beryllium) target with these gamma-rays, Cockroft, Duckworth, and Merrison (C16) obtained w Ta p\ 3u 6 3 W 11 12 13 14 15 16 17 18 19 20 21 7 -ray energy (Mev) Fig. 30b. (y,n) cross sections vs. 7-ray energy for copper, silver, and tantalum (D15.M13). average neutron yields of ~10 9 neutrons/sec and peak yields of ~2 X 10 12 neutrons/sec (in ~2 ^sec bursts). Calculations based on the curves of Fig. 30 indicate that an electron beam of 1 /tamp (average) and 20 Mev energy, impinging directly on a thick target of a medium or heavy element, should yield ~10 10 neutrons/sec. The spectrum of neutron energies resulting from x-ray-induced (y,n) reactions is relatively broad, corresponding to "evaporation" from an excited compound nucleus; the theory of the evaporation process, and of its Z-dependence, has been discussed by Heidmann and Bethe (H52). The Maxwellian form of the neutron energy distribution has been veri- fied by Byerly and Stephens (B80). However, there is expected to be a high-energy "tail" of neutrons ejected by direct gamma-ray action, with a non-spherical angular distribution (C29). 374 The Neutron [Pt. VII 3. Accelerated Charged Particle Sources. Radioactive alpha-particle and gamma-ray neutron sources are of rather limited usefulness, espe- cially for the production of intense beams of monoenergetic neutrons. The availability of Van de Graaff and cyclotron accelerators, capable of delivering strong monoenergetic currents of various nuclear projec- tiles, makes possible the production of strong neutron sources by a variety of nuclear reactions. These projectiles include protons (p), deuterons (d), tritons (t), alpha-particles (a), and heavier nuclei. In this section we shall consider only reactions induced by the first three. [(a,n) reactions have been discussed in connection with radioactive neutron sources.] We shall mainly emphasize, in this section, reactions which can lead to monoenergetic neutrons. This possibility exists whenever the energy of the first excited state of the product nucleus is too great to be excited by the projectiles used. Depending on the nucleus involved, the first excited state may lie anywhere from a few hundred kev to many Mev above the ground state. In some cases, sources may be effectively mono- energetic even when the first excited state is energetically available, owing to a relatively low yield of excited product nuclei. With the available reactions which yield monoenergetic neutrons, it appears possible to cover the neutron energy range from a few kev to 20 Mev. To cover the rest of the very fast- and the ultrafast-neutron ranges, it is necessary to resort to heteroenergetic sources, and to de- pend on the detector for any sorting out of the neutron energies. Prior to the availability of tritium, and the use of the t-d reaction, heteroener- getic (d,n) reactions were the only ones available for obtaining very fast neutrons. Such sources will also be considered in this discussion. Given the requisite reaction, a strong current of monoenergetic projectiles, and a thin target, the neutrons emerging at a given angle with respect to the projectile direction have a definite energy. The rela- tionship between the neutron energy E n , the angle 8, the projectile energy E i} the reaction Q value, and the masses of the particles involved has been frequently described. 1 The properties of monoenergetic neu- tron sources are collected in a review article by Hanson, Taschek, and Williams (H34), upon which this summary leans heavily. The energy- angle relationships in a given reaction are conveniently represented in the form of a nomograph, developed by McKibben (M15), 2 of which some examples will be shown in the following. 1 See, for example, Part VI. 2 Such nomographs for the most extensively used reactions can be purchased as document MDDC 223 from the Document Division of the AEC, Oak Ridge, Ten- nessee. They have been reprinted in the article of Hanson, Taschek, and Williams (H34). Sec. 3B] Sources and Detectors 375 A McKibben nomograph (Figs. 31, 33, 37, 40) consists of two sets of semicircles (solid and broken) and two sets of radial lines. The solid semicircles, centered at the origin of the (lower) neutron energy scale, represent various values of the neutron energy in the laboratory coordi- nate system. The solid radial lines, from the same origin, represent angles of emission of the neutrons in the laboratory system. The broken semicircles (which are not concentric) represent various energies of bombarding particle (p or d) ; the bombarding energy is given by the intersection of a broken semicircle with the upper of the two horizontal axes. The broken radial lines represent loci of equal values of the angle of neutron emission in the center-of-mass coordinate system. Thus, given a value of the bombarding particle energy, the energies of the emitted neutrons are given by the intersections of the appropriate broken semicircle and the solid semicircles; to each neutron energy (in- tersection) there corresponds a definite laboratory angle (solid line) and a definite center-of-mass angle (broken line). (a) Intermediate and Fast Neutrons from (p,n) Reactions. These reac- tions are endoergic. Hence, by bombarding thin targets with protons of energy only slightly above threshold it is possible to obtain monoener- getic neutrons of relatively low energy. The minimum energy obtain- able from such sources is limited by the fact that, at proton energies only slightly above threshold, there are two neutron energies correspond- ing to each angle of emergence. (Neutrons emitted at forward and back- ward angles in the center-of-mass system all lie within a cone of apex angle <180° in the laboratory system.) However, as soon as the apex angle of the cone of neutron emergence becomes 180°, the energy-angle relationship is unique. In this respect, the heavier the target nucleus, the lower the energy at which the neutrons for a given angle are mono- energetic. However, the necessity for penetration, by the proton, of the Coulomb barrier limits the possible target nuclei to low Z (<25). Some properties of known {p,n) reactions are summarized in Table 19. Most of these data are from the work of Richards and co-workers at the University of Wisconsin, and have been collected by Richards, Smith, and Browne (RIO), who give references to the original investigations. The minimum neutron energy at threshold (fifth column) arises from the center-of-mass motion of the system. E t E - mi ° = CA+I) 5 (86) where A is the mass number of the target nucleus. The sixth column gives the minimum energy of monoenergetic neutrons in the forward 376 The Neutron [Pt. VII TABLE 19 Properties op (p,n) Reactions for Z < 25 (Mostly from RIO) Target Product Observed E, (Mev) -Q (Mev) ■^w.min (kev) -^ re.min at0° (kev) Lowest Level (Mev) xH 2 21H 1 3.339 ±0.015 2.225 371 1979 iH« 2 He 3 1.019 ±0.001 0.764 63.7 286.5 >2.5 3 Li 7 4 Be 7 1.882 ±0.002 1.646 29.4 120.1 0.435 4 Be 9 5B 9 2.059 ±0.002 1.852 20.6 83.4 >1.5 6 B" eC 11 3.015 ±0.003 2.762 20.9 84.5 2.02 6 C 12 6 N 12 20.0 ±0.1 18.5 118 477 eC 13 6 N 13 3.236 ± 0.003 3.003 16.5 66.4 2.383 6 C 14 6 N" 0.664 ±0.009 0.620 2.9 11.8 2.3 8 18 8 F 18 2.590 ±0.004 2.453 7.2 28.8 9 F 19 ioNe 19 4.18 ±0.25 3.97 10.5 42 nNa 23 i 2 Mg 23 4.78 ±0.3 4.58 8.3 33 itCI" isA 37 1.640 ±0.004 1.598 1.1 4.6 1.4 18 A«> 19K* <2.4(?) 2.3(?) 1.6 5.7 0.81 i»K« 2oCa 41 1.25 ±0.02 1.22 0.7 2.8 1.95 2lSc 45 22 Ti 45 ~2.85 2.79 1.35 5.4 2 3 V 61 2 4Cr» 1.562 ±0.006 1.532 0.58 2.3 0.775 f 25 Mn 55 26 Fe 66 1.18 ±0.01 1.16 0.38 1.5 1.020 ±0.010 1.00 0.33 1.3 0.42 % t (S52). t (S54, M10). direction, E' ntm - m , which corresponds to the forward cone just filling the forward hemisphere. The properties of the most important of these neutron sources follow: H 3 (p,n)He 3 McKibben's nomograph for this reaction is shown in Fig. 31. Since tritium has only recently become extensively available, this reaction has not been very widely used for a neutron source; only the Wisconsin group has reported extensive (cross section) measurements with this source (B41, M31). Most of the information concerning yields, angular distributions, etc., vs. proton energies up to 2.5 Mev is due to the Los Alamos group (J2, H34). The cross section is shown in Fig. 32. The rapid increase may be due to a resonance of He 4 corresponding to in- cident protons of >2.5 Mev energy. The angular distribution of the emerging neutrons is quite complex (J2). A large yield of 20-Mev T(p,re)He 3 3.0 180^- E n (Mev) Fig. 31. McKibben nomograph of energy relationships in the H 3 (p,n) He 3 reaction (H34). Directions for its use are given on p. 375. 'o e J 30 *? 0.10 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 •Ep(Mev) Fig. 32. Total cross section of the H 3 (p,n)He 3 reaction as a function of the incident proton energy {32). 377 378 The Neutron [Pt. VII 4.0 3,5 3.0 2.5 ,iiiii — r 1.0 0.5 2.0 E p 1.88 (Mev) 2.0 3.0 3.5 4.0 ! I I I i I I M I I | i I I ' I" 0.5 1.0 1.5 2.0 E n (MeW Fig. 33. McKibben nomograph for the Li 7 (p,ra)Be 7 reaction (H34). Directions for its use are given on p. 375. 11 10 - I / \ F-40-kev Li target 4 ' J ■ / ■ . ' 7 ''E n (0°) ' ' \ . / \ / / 4 1 / \ j / \ / [/_ } J / 1/ 1 } f / 2.0 s 1-5 X 1.0- 0.5 1.8 2.0 2.2 2.4 2.G 2.8 3.0 3.2 3.4 3.6 Proton energy (Mev) Fig. 34. Yield of the Li 7 (p,«)Be 7 reaction in the forward direction (H3-4). Sec. 3B] Sources and Detectors 379 gamma-radiation, arising from the H 3 (p,7)He 4 reaction, accompanies the (p,n) reaction, and may interfere with some forms of neutron detec- tion (A30). Li 7 ( P ,n)Be 7 This reaction has provided the most extensively used source of inter- mediate-energy neutrons. Its nomograph is shown in Fig. 33. The S 0.30 0.20 L i 7 (p, «)B 3 7 cross section as a function of energry' - 2 Points from Be 7 activity ^1 Calibration points from Mn 55 bath standardizatio o Points from integrated angular distributions i Foir its I rom E1SS1C n ra tes t\ / : : \ c Y : \ c 1 A i J ( t . ' ■> ■* J — ~»— — Extrapolation to threshold assumed linear in E„ £n£±£* Extrapolation to threshold assumed linear vas/E^ i i 1.9 2.0 2.1 2.2 2.3 Proton energy (Mev) 2.4 Fig. 35. Total cross section for the Li 7 (p,ra)Be 7 reaction (H34). yield in the forward direction is shown in Fig. 34. The first peak is due to the concentration of neutrons in the forward cone for proton energies close to threshold. The second peak, at E p « 2.3 Mev, is real, and corresponds to a resonance of the compound nucleus Be 8 . The angular distribution is peaked in the forward direction, especially in the vicinity of the resonance, and requires terms up to cos 2 (P 2 ) for its description (Tl). The total cross section vs. proton energy is shown in Fig. 35. 380 The Neutron [Pt. VII Until relatively recently, it was thought that the neutrons from this reaction were truly monoenergetic (H8). However, the reaction is now known to yield a low-energy group of neutrons corresponding to an ex- cited state of Be 7 at 435 kev. The results of various investigators on the relative yield of the low-energy group are summarized in Table 20. TABLE 20 lTivb Yield of Neutrons Arising from Excitation of the 435-] State OF Be 7 IN THE Li 7 (p,m)Be 7 * Reaction Intensity Relative Proton Energy Angle of to Ground-State (Mev) Observation Group (%) References 2.378 (E t *) 0° ~3 W33 2.705 0° 8±2 H20 2.75 30° 9± 1.5 J5 2.89 30° 10.5 ± 1 J5 3.120 0° 8±2 H20 3.31 0° 17 ± 10 J8 3.49 0° 10 ±3 F42 3.66 30° 12 ± 1 J5 3.91 0° 9±4 J8 60° 16 ±6 J8 Additional Be 7 levels have been reported below 1 Mev (G30), but these have not been confirmed in other investigations (K6). Other Possibilities Other possible (p,n) reactions are discussed by Hanson, Taschek, and Williams (H34) and by Richards, Smith, and Browne (R10). The Be 9 (p,n)B 9 reaction is quite similar to that on Li 7 , but does not give a low-energy group up to neutron energies of ~1.5 Mev (J9). However, its yield is lower, and thin beryllium targets are considerably more difficult to prepare. The (p,n) reactions on scandium, vanadium, and manganese are being considered for producing monoenergetic neutrons in the ~l-kev energy range. Their yields have been studied by a number of investiga- tors (H34, R10, B4, M10, S52, S54). (6) Fast and Very Fast Neutrons from (d,ri) Reactions. These reac- tions are, with the exception of C 12 (d,n)N 13 , exoergic. They are useful in producing fast neutrons in the range 1-20 Mev. The Q values for the most useful (d,n) reactions are shown in Table 21. The last two yield heteroenergetic neutrons, but have been extensively used because of their high yields. Sec. 3B] Sources and Detectors 381 TABLE 21 Properties op Some (d,n) Reactions Reaction Q Value (Mev) H 2 (d,re)He 3 3.265 ±0.018 (T16) H 3 (rf,w)He 4 17.6 (H34) C 12 (d,w)N 13 -0.281 ± 0.003 (T16) N 14 (d,n)0 15 5.1 (H34) Li 7 (d,n)Be s 15.0 (T16) Be 9 (d,n)B 10 3.79 (T16) E n0 for E d = (Mev) 2.45 14.1 (E, = 0.328) 4.8 13.3 3.44 The last column gives the energy E n0 of neutrons which would result from the absorption of zero energy deuterons; this energy is independent 1 2 3 4 5 6 7 Energy of accelerated particle (Mev) Fig. 36. Energies of neutrons, emitted at 0° and 180°, for the most important mono- energetic (d,n) and (p,ri) reactions (H34). of the angle of emergence of the neutron. For deuterons with a given kinetic energy, the neutron energy is uniquely determined by its angle of emergence, provided that the reaction goes to the ground state of the product nucleus. The neutron energies vary on both sides of E n0 , being greater for emission in the forward direction and less for backward emission. The energies of neutrons emitted at 0° and 180° are shown in Fig. 36, as a function of the bombarding energy, for the first three reac- 382 The Neutron [Pt. VII tions in the table. Also included are curves for the two (p,ri) reactions discussed above. H 2 (d,n)He 3 The d-d reaction has been extensively used, and its characteristics widely investigated (H34). It has high yields for low bombarding energies, which makes it particularly useful as a neutron source with low-voltage (<1 Mev) particle accelerators. The neutrons are mono- energetic up to bombarding energies of 10 Mev. Figure 37 is the Mc- Kibben nomograph for this reaction. EV(Mev) 2 3 Fig. 37. McKibben nomograph for the d-d reaction (H34). Directions for its use are given on p. 375. The angular distribution of d-d neutrons has been extensively studied, both experimentally and theoretically. At energies of a few hundred kev, the distribution in the cm. system is symmetrical about 90°, being ~60 percent lower at 90° than at 0° and 180° (H34, M16). For higher energies the distribution becomes much more anisotropic. Hunter and Richards (H78) have investigated the angular distribution for deuterons of from 0.5 to 3.7 Mev. They find that it can be represented by an ex- pression of the form N(fi) = A P (d) + A 2 P 2 {6) + AiP^e) + A 6 P 6 (d) +■■■ (87) in which the P„(0)'s are the Legendre polynomials, and all the coeffi- cients vary with energy, as shown in Fig. 38a. The theory of the angular distribution has been discussed by Konopinski and Teller (K21) Sec. 3B] Sources and Detectors 383 and by Beiduk, Pruett, and Konopinski (B14). The total cross section vs. deuteron energy is shown in Fig. 38b. E d (Mev) Fig. 38a. Variation with deuteron energy, E d , of the coefficients of the Legendre polynomial fit to the angular distribution of the neutrons from the d-d reaction in the center-of-mass coordinate system (H78). ' ' ' i I r 0.10 o ^S^~* " "° ' "^ s— * • ^0.08 D / Symbol on Reference No. 01 o Figure in Legend S, 0.06 to - o* . • Q 1 2 3 0.04 - & .8 A 4 5 0.02 a - + E d (Mev) (1) Hunter and Richards (H78). (2) Blair, Freier, Lampi, Sleator, and Williams, Phys. Rev., 74, 1599 (1948). (3) Manley, Coon, and Graves, Phys. Rev., 70, 101(A) (1946). (4) Graves, Graves, Coon, and Manley, Phys. Rev., 70, 101(A) (1946). (5) Bretscher, French, and Seidl, Phys. Rev., 73, 815 (1948). Fig. 38b. Total cross section for the H 2 (d,n)He 3 reaction (H78). The d-d reaction is frequently used to provide strong neutron sources by bombarding thick targets of heavy ice with deuterons. The thick target yield curve (H34) is given in Fig. 39. 384 The Neutron [Pt. VII The companion reaction H 2 (d,p)K 3 has a comparable cross section (M16, H34). The protons are frequently used to "monitor" the neutron yield. At high bombarding deuteron energies the He 3 recoils from the 1 /2 10 'o 2 5 0.2 0.4 0.6 0.8 Deuteron energy (Mev) 1.0 Fig. 39. Thick heavy ice target H 2 (d,ra)He 3 neutron yields (H34). (d,n) reaction can be used as a monitor, and also, by observing He 3 neutron coincidences, to eliminate background effects due to (d,ri) re- actions on target impurities. H 3 (d,w)He 4 This, the t-d reaction, is the most strongly exoergic of the reactions capable of yielding monoenergetic neutrons; it can provide neutrons of from 12 to 20 Mev by using deuterons of up to 3 Mev (Fig. 36). The McKibben nomograph for the reaction is shown in Fig. 40. This reac- tion is especially useful with low-voltage deuteron accelerators, since it has very large yields for deuterons of a few hundred kev energy, owing to a resonance in the cross section at ~100 kev. The low-energy cross- section data (up to ~1 Mev deuterons) can be fitted by the expres- sion (H34) 58 exp (-1.72/2^) a = — — — — — barns (88) E 1 + {E - 0.096)7(0. 174)' where E is the deuteron energy in Mev. From 1 to 2.5 Mev the cross section remains essentially flat at ~0.15 barn. The cross section is 0.05 barn at 10.5 Mev (B72). Sec. 3B] Sources and Detectors 385 The angular distribution in the t-d reaction, for incident deuterons up to 2.5 Mev, has been measured by Taschek, Hemmendinger, and Jarvis (T2) by observing the distribution of recoil alpha-particles. Their re- T(d,m)He 4 10 20 E {Mev) 10 Pig. 40. MeKibben nomograph for the t-d reaction (H34). Directions for its use are given on p. 375. suits are shown in Fig. 41. Detection of the recoil alphas provides an effective means for monitoring the neutron yield from this source. The angular distribution for E d = 10.5 Mev has been measured by Brolley, Fowler, and Stovall (B72). ~0.016 "3 * 0.012 "0.008 ^-^- T fflEJ-- 2.5 | ^4-A°;f<4.2.5 1.5 — - ± ! 0.024 40 50 60 70 80 90 100 110 120 130 Laboratory angle (degress) 30 60 90 120 150 Center of Gravity angle (degrees) Fig. 41. Angular distribution of recoil alpha-particles from the H 3 (<2,n)He 4 reac- tion for various deuteron energies (in Mev) (T2, H33). 386 The Neutron [Pt. VII Because of the strong low-energy resonance, the t-d reaction is expected to have large thick target yields. Thus, for a thick gas target and 600- kev d's the yield is 5 X 10 8 neutrons per microcoulomb ; for tritons ad- sorbed in a thick zirconium target, the yield at Ed = 200 kev is about 10 8 neutrons per microcoulomb (H34). One interesting application of the t-d reaction is in the possibility of converting intense sources of thermal neutrons (such as those available from piles) into strong fast-neutron sources. Thus, if thermal neutrons are allowed to fall on a mixture of lithium and deuterium, fast tritons (E = 2.65 Mev) will be produced by the Li 6 (n,a)H 3 reaction. These tritons then react with the deuterons to produce 14-Mev neutrons by the t-d reaction. A recent investigation by Almqvist (All) has shown that such sources also produce neutrons by reactions of tritons on the lithium nuclei; the latter reactions have a considerably greater yield than the t-d reaction in the mixtures used. Since the neutrons from lithium result from a variety of reactions on both lithium isotopes, with Q values ranging from 16 to 8.9 Mev, the neutrons from such sources have a complex spectrum. The neutron yields for a number of com- pounds are shown in Table 22. From these data the author derives TABLE 22 Neutron Yields fkom Thermal Neutron Irradiation of Lithium and Lithium-Deuterium Compounds (All) Neutrons per 10 6 Tritons Substance From Li From '. LiF 26.8 Li 2 C0 3 17.2 LiOH-H 2 15.2 LiOD 20.3 3.4 LiODD 2 + D 2 9.3 12.1 average cross sections of 1.5 and 0.41 barns for 2.65-Mev tritons on thick targets of lithium and deuterium, respectively. Crews (C32) has measured the yield and angular distribution of neutrons from thin lithium targets for triton energies between 0.25 and 2.10 Mev; he derives a ~ 0.76 barn for E t = 2.0 Mev on a thick Li target. C 12 (rf,n)N 13 This, among the (d,n) reactions of interest, is the only endoergic reaction. It is mainly useful because of its low threshold (Table 21; Sec. 3B] Sources and Detectors 387 Fig. 36), which makes it a convenient low-voltage accelerator source for neutrons of from a few hundred kev to ~1.5 Mev. Since the first ex- cited state of N 13 , observed in this reaction, is at ~2.3 Mev (G29), the neutrons are monoenergetic to —2 Mev. The reaction yield is rela- tively low and shows a complex angular distribution and energy de- pendence (H34). The companion reaction C 12 (d,p)C 13 has a similar yield curve at low deuteron energies, and is useful for monitoring the neutron yield. A complicating feature is the presence, with normal carbon targets, of the reaction C 13 (d,n)N 14 , with a Q value of 5.2 Mev, which gives a fast- neutron group of intensity <~1 per cent of the C 12 neutrons. N 14 (<Z,n)0 15 This reaction, very seldom used, gives monoenergetic neutrons for deuterons of up to ~1 Mev energy. At 1 Mev the cross section relative to that of the d-d reaction has been measured by Gibson and Livesey (G4), and is shown in Table 23. TABLE 23 Ratio of d-d to N 14 -rf Cross Sections for 1-Mev Deuterons (G4) Angle Ratio (d-d/N u -d) 0° 4.4 30° 4.4 90° 1.5 150° 2.1 U 7 (d,n)Be 8 Because of its large Q value and large yield, this source has been exten- sively used, especially with cyclotrons. However, the large number of levels, in both the compound and product nuclei, gives rise to a com- plicated neutron spectrum with a wide energy spread (H34). The thick target yield for a bombarding energy of 600 kev is 17 X 10 6 neutrons per microcoulomb. Above E d = 1 Mev the thick target yields are greater than for the d-d reaction. Be 9 (d,w)B 10 This reaction also gives rise to a complex neutron spectrum of lower maximum energy than that from Li 7 (A4). The thick target yields at 388 The Neutron [Pt. VII low deuteron energies are shown in Table 24. Above ~1 Mev the thick target yields are between those of the d-d and the Li 7 (d,ri) reactions. TABLE 24 Thick Taeget Yields fob the Be 9 (d,w)B 10 Reaction Ed (kev) 10 6 Neutrons/Microcoulomb 400 4 600 21 800 106 Other Possibilities Many other (d,n) reactions have been employed as neutron sources. Among these the F 19 (d,ra)Ne 20 , Q = 10.7 Mev, has received some atten- tion, but its characteristics are not more favorable than the other reac- tions which yield heteroenergetic sources, and its yield is lower. While such sources have frequently been used in the past to produce very fast neutrons, they have been mainly superseded by the t-d reaction. The (d,ri) yields of a variety of elements, for deuteron energies of ~10 Mev and greater, have been studied by a number of investigators. Thick target yields for 10-Mev deuterons (due to Smith and Kruger, S41) are shown in Table 25, together with the yields at 15 Mev (due to Allen, Nechaj, Sun, and Jennings, A8). The latter results are sur- TABLE 25 Neutron Yields from 10- and 15-Mev Deuterons on Thick Targets Yield (10 8 neutrons/ Yield (10 s neutrons/ microcoulomb) microcoulomb) Target Z Target Z 10 Mev 15 Mev 10 Mev 15 Mev Be 4 320 190 Nb 41 15 B 5 190 Mo 42 42 15 C 6 120 Ag 47 14 Al 13 87 64 Cd 48 12 P 15 105 Sb 51 35 Ti 22 65 Ta 73 7.4 3.3 Cr 24 29 W 74 7.0 Mn 25 76 52 Pt 78 6.0 Co 27 26 Au 79 4.7 2.1 Ni 28 33 Pb 82 2.1 Cu 29 55 29 Bi 83 1.3 Sec. 3B] Sources and Detectors 389 prisingly lower than those at 10 Mev, although the trend with atomic number is the same for both series of measurements. The discrepancy probably arises, at least in part, from the fact that the 10-Mev measure- ments counted neutrons of all energies while the 15-Mev results apply only to those neutrons capable of exciting the S 32 (n,p) reaction. The angular distributions of (d,n) neutrons have been investigated by Roberts and Abelson (R14), by Falk, Creutz, and Seitz (Fl), by Allen et al. (A8, S58) (all at E d = 15 Mev), by Ammiraju (A22) {E d = 18 Mev), by Schecter (S4) {E d = 20 Mev), and by others. These distri- butions show a strong peaking in the forward direction. [Some, e.g., Be, when only the highest-energy neutrons are detected, show structure in the angular distribution (S58) .] The observed characteristics of the (d,ri) reaction for high-energy deuterons can be explained in terms of the "stripping" theory of Serber (S20) and Peaslee (P8), according to which most of the neutrons are produced by a process in which the neutron never enters a compound nucleus; the deuteron is polarized in the field of the nucleus and then split, the proton being captured (and occasionally scattered) by the tar- get nucleus. Recently, Butler (B78, B79) has made a significant con- tribution to the "stripping" theory by considering in greater detail the angular distribution of neutrons from such reactions when the product nucleus is left in a definite quantum state. His calculations predict structure (i.e., maxima and minima) in the angular distribution which can be interpreted in terms of the angular momentum and parity properties of the initial and final states. (c) Ultrafast Neutrons from Accelerated Deuterons and Protons. When charged particles from ultrahigh-energy accelerators strike a target, neutrons are produced through a variety of reactions. Those reactions which involve the capture of the bombarding particle into a compound nucleus give rise to neutrons by the process of evaporation. Such neu- trons have the broad energy distribution characteristic of the statistical theory, with the maximum in the fast- or very fast-energy region and a more-or-less symmetrical angular distribution. However, when deu- terons (and, to a lesser extent, protons) are used as the bombarding particles, there is observed, superimposed on the evaporated neutrons, a strong forward peak of ultrafast neutrons with a relatively narrow energy distribution. In the case of deuteron bombardment, the neu- trons arise mainly from the stripping process, mentioned briefly in the preceding section. Serber (S20) has given a simple picture of this process, as follows: The deuteron is a relatively loosely bound structure in which neutron and proton spend a large fraction of the time far apart. Hence, it is 390 The Neutron [Pt. VII not improbable that, as a deuteron traverses the target, the proton will strike one of the target nuclei while the neutron remains outside the nucleus. In such a collision the proton will be stopped while the neu- tron will go on, carrying off approximately half of the original deuteron energy. However, the neutron has internal motion, relative to the center of mass of the deuteron. Hence, as the neutron leaves the proton behind, this relative motion is superimposed on the forward motion of the deuteron, giving rise to an energy spread centered around the average value of half the deuteron energy. Another result of the relative motion is that the emergent "stripped" neutrons have a spread in directions, around the original deuteron direction. The magnitudes of the energy and angular spreads of the neutron beam can be estimated as follows: The forward momentum of the neutron, due to the kinetic energy, E d , of the deuteron, is Po = (ME d ) 1A (89a) The internal momentum of the neutron is, roughly, Vi ^ (Me d ) y ° (89b) where t d is the deuteron binding energy. The stripped neutrons have energies within the limits given by (po ± Pi) 2 1 E ^ n P ~2 Ed± {tdEd) (90) A more accurate calculation (S20) gives, for the full width at half maximum, AE = 1.5(tdE d ) H (90a) The angular spread in the neutron beam is determined by the relative values of the forward and transverse neutron momenta: A0 « =* 2—1 radians (91) Po \eJ The more accurate calculation (S20) gives A0 = 1.6 (|-) ' (91a) as the full angular width at half maximum. The above considerations, with respect to the angular spread, apply only to targets of light nuclei, in which the coulomb deflection of the deuterons is negligible. For the Sec. 3B] Sources and Detectors 391 heaviest nuclei the coulomb deflections lead to about twice as large a spread as Eq. (91a). The energy and angular distributions of the stripped neutrons are superimposed on the background of neutrons produced as a result of evaporations. However, this background is expected to be small, especially since it is spread, approximately uniformly, over all direc- tions. In addition, some neutrons are produced by the disintegration of deuterons in the coulomb field of the target nuclei; this effect is rela- tively small for light nuclei and becomes comparable to the stripping production for the heaviest target materials (Dl, K20). 40 60 80 100 Neutron energy (Mev) 120 140 Pig. 42. Energy distribution of neutrons obtained by stripping 190-Mev deuterons in a 1.27-cm thick beryllium target, due to Hadley et al. (HI). The curve is from the theory of Serber (S20). The observations of Helmholz, McMillan, and Sewell (H53) on angular distributions of the neutrons produced by bombarding various nuclei with 190-Mev deuterons and of Hadley, Kelly, Leith, Segre, Wiegand, and York (HI) on the energy distribution of the neutrons from a beryllium target bombarded by 190-Mev deuterons are in good agreement with Serber's theory, and demonstrate the possibility of using the stripping process to obtain relatively monoenergetic ultra- fast neutron beams from ultrahigh-energy particle accelerators. The energy distribution due to the latter investigators is shown in Fig. 42. When ultrafast protons are used as the bombarding particles, the emerging neutrons are neither so sharply collimated nor so nearly mono- energetic. However, observations by Miller, Sewell, and Wright (M32) (330-Mev protons on beryllium, aluminum, copper, and uranium) and by DeJuren (Dll) (340-Mev protons on beryllium) have demonstrated that there is an appreciable component of ultrafast neutrons emitted at forward angles. The results of the first-mentioned investigators, on the full angular widths at half maximum for neutrons of E > 20 Mev, are: beryllium, 54°; aluminum, 59°; copper, 59°; uranium, 58°. 392 The Neutron [Pt. VII Such forward-collimated, ultrafast neutrons result primarily from one or a few p-n collisions in the nucleus ; at these energies the scattering of neutrons by protons is predominantly forward (charge exchange), and the scattered neutron has an appreciable probability of emerging from the nucleus without undergoing any further scattering. Relative neutron yields for targets of various Z have been measured by Knox (K20) with both high-energy deuterons and protons as the bombarding particles. For 190-Mev deuteron bombardment the rela- tive yields agree well with stripping plus disintegration by the Coulomb field. In the case of 340-Mev proton bombardment, the yield varies approximately as (A — Zf 3 . The energy distribution of the neutrons from beryllium and carbon targets bombarded with ~100-Mev protons has been investigated by Bodansky and Ramsey (B42, B43). For a beryllium target they ob- serve a sharp peak at E n « 93 Mev, with a width at half maximum of ~30 Mev. Below ~70 Mev, the neutron energy distribution becomes essentially fiat, with a yield of ~40 percent of the peak value. The neu- tron yield from a carbon target shows a slight peak at ~70 Mev (~25 percent above the roughly constant yield below E n = 60 Mev) and falls off rapidly above 70 Mev. Similar studies on neutrons from beryl- lium, carbon, aluminum, and uranium bombarded with 170-Mev pro- tons have been made by Taylor, Pickavance, Cassels, and Randle (T3). 4. Neutrons from U 235 Fission. One of the most important present- day neutron sources is the fission reaction. Fission of the heaviest ele- ments is accompanied by the emission of fast neutrons. The neutron spectrum resulting from thermal neutron fission of U 235 is closely approximated by the expression (U2) <W — • = e-^sinh(2£/) M (92) dE where E is the neutron energy in Mev. This expression, suggested by Watt, agrees with the observed fission spectrum to within ±15 per- cent up to E = 17 Mev. 1 Equation (92) is plotted in Fig. 43. As seen in the figure, the fission spectrum has a rather broad energy distribution, with an average energy of ~1.5 Mev. However, this is by no means the spectrum of neutrons normally observed either inside or emerging from piles, since neutrons in piles suffer considerable modera- tion by elastic scattering on light nuclei or by inelastic scattering on heavy nuclei. Nevertheless, strong sources of unmoderated fission neu- 1 See T. W. Bonner, R. A. Ferrell, and M. C. Rinehart, Phys. Rev., 87, 1032 (1952); D. L. Hill, Phys. Rev., 87, 1034 (1952); B. E. Watt, Phys. Rev., 87, 1037 (1952). Sec. 3B] Sources and Detectors 393 trons can be obtained by irradiating a uranium target with an intense thermal neutron beam (as from a pile). Such a source was used in the measurements by Hughes et al. (H72) of capture cross sections for fast neutrons, discussed in Section 2. 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.1 1 2 3 4 5 6 7 8 9 10 11 12 13 E n (Mev) Fig. 43. Approximate shape (to within ±15%) of the neutron spectrum from the fission of U 236 by thermal neutrons (U2). 5. Neutrons in the Cosmic Radiation. Because the neutron is un- stable, it is not expected to be one of the components of the primary cosmic radiation. Nevertheless, neutrons constitute an appreciable frac- tion of the "hard" or "penetrating" component of cosmic rays. These neutrons arise from nuclear disintegrations (stars) in the atmosphere and are responsible, in turn, for a large fraction of the observed nuclear disintegrations, especially at low altitudes (R21, S38, K22). The observations on neutrons in the cosmic radiation are of two general types: (1) The detection of very fast and ultrafast neutrons by observation of their interaction with nuclei, especially star production. Of such experiments, the most unambiguous are those in which stars are observed in sensitive nuclear emulsions, where the particle responsible for the disintegration can be seen or, in the case of neutrons, deduced from the absence of any charged particle of sufficient energy. Results of such experiments are reported and summarized by Bernardini, Cortini, and Manfredini (B18). 394 The Neutron [Pt. VII (2) The detection of slow neutrons in paraffin-surrounded and cadmium-covered BF 3 counters (Y4, S50). Many such experiments have been performed by many investigators; one of the most recent and most extensive, due to Yuan (Y4), has been carried to an altitude of ~100,000 ft (pressure ~1 cm Hg). The thermal neutron intensity shows an exponential increase (mean free path ~ 156 g/cm 2 ) up to about 20 cm of Hg, a maximum at ~8.5 cm of Hg, and a rapid falling off at higher altitudes. The interpretation of experiments of the second type requires con- sideration of the processes by which the neutrons, produced mainly as fast neutrons in nuclear disintegrations, lose energy by scattering and, at the same time, diffuse through the atmosphere. (The general dis- cussion of such problems is reserved for Section 4.) The theory of the slowing down of neutrons in the atmosphere has been given by Bethe, Korff , and Placzek (B26) ; the diffusion problem is discussed by Flugge (F35). Their calculations have recently been revised (in the light of more accurate data on nuclear cross sections) by Davis (D5) and by Lattimore (L6). 6. Sources of Slow Neutrons. By far the greatest number of neutron investigations have involved the use of slow neutrons. Sources of slow neutrons are invariably derived from intermediate- or higher-energy neutrons by allowing the neutrons to diffuse through a "moderating" material. [The lower limit of monoenergetic neutrons from threshold charged particle reactions, mostly (p,n), is being slowly pushed toward the slow-neutron region.] Neutrons lose energy through inelastic collisions with medium or heavy nuclei and through elastic collisions with light nuclei; the first process is most effective for fast neutrons and the second for interme- diate and slow neutrons. Consider a source of fast neutrons embedded in a large mass of material, preferably one containing light nuclei. The neutrons diffuse through the material, losing energy as they suffer scat- terings. If the capture cross section is small compared to the scattering cross section (as it is in the intermediate- and fast-neutron ranges for most materials, especially light elements), the neutrons continue to lose energy until their energy is comparable to the energy of thermal agita- tion of the nuclei in the moderating material. After this, a neutron is as likely to gain as to lose energy in a scattering, and the slowing down stops. The neutrons are then said to be in "thermal equilibrium" with the moderating medium. The energy distribution of neutrons in thermal equilibrium is in many respects similar to that of the molecules of a gas at the temperature of Sec. 3B] Sources and Detectors 395 the medium. It is closely approximated by the Maxwell distribution function, dn — = Av 2 e- M * 2 /M T (93) dv where M is the neutron mass, k is Boltzmann's constant, and T is the absolute temperature of the moderating medium; the factor A is gen- erally a function of the position in the medium. The Maxwell distribu- tion gives as the most probable velocity o'-i^) (94) /2kT\ \W which is 2.2 X 10 5 cm/sec for T = 300°K; the average velocity is 2 v = —v' = 1.128?/ (95) Strictly speaking, in an infinite homogeneous moderating medium the thermal neutrons will have a Maxwell energy distribution only under special circumstances, of which two are: (1) the nuclei in the moderating medium scatter but do not absorb neutrons; (2) the neutrons are emitted into the medium with the Maxwell energy distribution and the absorp- tion cross section obeys the 1/v law. In this case, since the rate of neutron absorption, ~nw(v), is proportional to n, the original distribu- tion shape will maintain itself. Most thermal neutron sources deviate somewhat from the Maxwell distribution. If the medium strongly absorbs neutrons in the thermal energy range, the neutrons will never attain very low velocities, and the average velocity will be greater than that given by Eq. (95). If the size of the slowing-down medium is not sufficiently great, the neutrons will escape before attaining thermal equilibrium. In this situation, the average energy will again be greater than in Eq. (95). In any event, there is always present in the medium, at distances relatively close to the source, a component of neutrons in the process of being slowed down. These have an energy distribution characterized by equal fluxes (<£ = nv) of neutrons in equal logarithmic energy intervals, i.e., BdE d<j> = — (96) where B depends both on the position in the medium and on the neutron energy. Such neutrons (called epithermal) are observed as a tail on the Maxwell distribution. 396 The Neutron [Pt. VII The theory of slowing down and diffusion of neutrons will be discussed in Section 4. Some results, useful in the planning and interpretation of experiments involving slow-neutron sources, are summarized below. The most frequently used moderating media are water or paraffin, and graphite. The former have the advantage of containing a large propor- tion of hydrogen, the most effective nuclei for slowing down intermediate energy neutron; hence, for a given source strength, higher thermal neutron densities can be obtained with smaller quantities of moderating material than for any other slowing-down medium. On the other hand, carbon has a considerably smaller capture cross section than hydrogen, so that it is possible to obtain purer thermal neutron sources by making observations at greater distances from the fast-neutron source. Other materials which have proved useful as slowing-down media include heavy water, beryllium, and beryllium oxide (BeO); these are inter- mediate between water and graphite in slowing-down effectiveness, but are considerably more difficult to obtain in sufficient quantity. Since the early work of Fermi and co-workers (F17, A20) on the moderating properties of water there have been many investigations of slow-neutron distributions in water for many fast-neutron sources. The results of one such study, due to Anderson, Koontz, and Roberts, are shown in Fig. 44, a plot of activity X r 2 (proportional to r 2 nv) vs. dis- tance from the source. The characteristic features of such curves are the peak at a relatively small distance and the exponential decrease at large distances from the source. The results of such studies are frequently summarized in terms of the value of the second moment of the distribution, | Ar 4 dr -* ° - (97) Ar 2 dr where A is the measured activity of the detector used. The measure- ments of Anderson, Koontz, and Roberts (Fig. 44) give r 2 = 278 and 353 cm 2 for the indium resonance (1.44 ev) and thermal neutron detec- tor, respectively. (These are in fair agreement with the results of other investigators.) The characteristic lengths of the exponential decrease at large distances, r 2 A(r)~e- rl * (98) were found to be X = 9.43 and 10.00 cm for the indium resonance and thermal detectors, respectively. Sec. 3B] Sources and Detectors 397 The slowing down of neutrons in graphite has also been studied with various sources. The results of measurements by Feld and Fermi are shown in Fig. 45 as curves of thermal and indium resonance detector 2.0 1.6 0.2 Tr ( « K / / j r\ / / 1 ierrr al n BUtr ons x 1( -' 11 / 1 / / 1/ 1 1 / 1 1 1 // '/ (1 / l\ Nj 1 12 16 20 24 Distance from source (cm) 28 Fig. 44. Distribution of slow neutrons from a Ra-a-Be source in water. The ordi- nate is the activity X r 2 of a thin indium foil. The curve labeled "indium reso- nance" represents the activity of a cadmium-covered foil; the curve labeled "thermal neutrons" represents the activity of a bare foil minus 1.07 times the activity of the cadmium-covered foil. The source had a strength of 13.2 X 10 6 neutrons/sec. The indium foils had an area of 3 X 6.4 cm 2 and a thickness of 1.69 g/cm 2 . The cadmium covers were 0.96 g/cm 2 thick. Data due to Anderson, Koontz, and Roberts (unpublished). The ordinate scale can be converted to a scale of thermal neutron flux (hv) X r 1 by multiplying by 1.6 X 10 6 . activation vs. distance from the source, for a Ra-a-Be source (Fig. 45a) and a Ra-7-Be source (Fig. 45b). Since these measurements were made in a graphite column of non-negligible lateral dimensions, the geometry- is not strictly comparable to that for a point source in an effectively infinite medium (Fig. 44). However, the geometric effects can be taken into account in interpreting such experiments (see Section 4). 398 The Neutron Ra-a-Be [Pt. VII V 180 \\ \ 160 V — Tl erm il 1 \ \\ 120 100 \ 80 \ \ \ \ •\ I res idiur onar n — - ce " > • s 120 Z (cm) Fig. 45a. Thermal neutron flux and indium resonance neutron flux for a Ra-a-Be source in graphite, density 1.6 g/cm 3 , due to Feld and Fermi (unpublished). All the data are normalized to the same value at z = 10.2 cm. The neutrons were slowed down in a rectangular graphite column, 4 ft by 4 ft square by 7 ft high. The source was placed on the long axis 2 ft from the bottom face of the column. The abscissa z is distance, along the column axis, between source and indium detector. The actual ratio of thermal to indium resonance activity at z = 10.2 cm for the detector used was 3.72. For a monoenergetic source in a graphite column, the distribution of indium resonance neutrons is expected to be Gaussian: Wm» = ae~ ( * /zo)2 (99) The distribution of indium resonance neutrons for the Ra-a-Be and Ra-Y-Be sources could be expressed as a superposition of 3 and 2 Gaus- sian functions, respectively, with the proportions and ranges given in Table 26. Sec. 3B] 200 Sources and Detectors Ra-y-Be 399 80 20 K \ \ V -Th ermj ii \ \ I \ \ \ \ ' ^ • Indium resonance 60 80 Z (cm) 100 140 Fig. 45b. Thermal neutron flux and indium resonance neutron flux for a Ra-7-Be source in a graphite column (see legend of Fig. 45a), due to Feld and Fermi (unpub- lished). The actual ratio of thermal to indium resonance activity at 2 = 10.2 cm for the detector used was 2.57. The ratio of the strength of the Ra-a-Be source (see Fig. 45a) to the Ra-7-Be source was 0.231. The corresponding ratio of thermal neutron activities at 10.2 cm was 0.355. The distribution of thermal neutrons is a somewhat more complicated function of distance from the source. However, at distances greater than ~60 cm, for the Ra-a-Be source, and ~30 cm, for the Ra-y-Be source, the thermal neutron density becomes exponential : (nv) th = 0e-* /x (99a) with the same value of X = 24.2 cm applying to both sources. It is the difference between the exponential (Eq. 99a) and Gaussian (Eq. 99) functions which results, in graphite, in the rapid increase of the ratio of 400 The Neutron [Pt. VII TABLE 26 Analysis of Indium Resonance Neutbon Distributions along the Axis op a Graphite Column fob Ra-a-Be and Ra-7-Be Sources in Terms of Gaussian Functions (Eq. 99 and Fig. 45) Ra-a-Be Ra-7-Be «» (%) «o (cm) oci (%) z (cm) 15.0 22.8 38.7 22 69.3 36.9 61.3 29 15.7 57.1 meanf 39.2 meant 26.5 t The mean is computed as the square root of the weighted mean square. thermal to epithermal neutron flux with increasing distance from the source. While slow-neutron sources of the types described above have been and continue to be extensively used, the most potent available slow- neutron sources are those associated with chain reactors (piles). The spectrum of neutrons inside a slow-neutron reactor contains an appre- ciable component of fast, intermediate, and epithermal neutrons. 1 How- ever, by use of a "thermal column," it is possible to obtain high fluxes of practically pure thermal neutrons. The usual thermal column is simply a large block of graphite placed directly adjacent to the pile. Neutrons which enter the column are slowed down as they diffuse to the outside, the ratio of epithermal to thermal neutrons falling off rapidly with the distance into the column. Figure 46 shows the activity of bare and cadmium-covered indium foils at various positions in the thermal column adjacent to the Argonne heavy-water-moderated pile. Thermal neutron fluxes, available from various sources, are summar- ized in Table 27. For many purposes, the thermal neutrons diffusing in a mass of moderating material can be utilized directly, in situ. Thus, for example, in irradiating a sample for the purpose of observing the resulting radio- activity, the maximum slow-neutron flux and consequently the maxi- mum activation are attained when the sample is embedded in the moderator as close as possible to the source. For some investigations, on the other hand, it is desirable to have a beam of thermal neutrons, e.g., for the measurement of cross sections by a transmission experiment. One method of obtaining a beam of thermal neutrons is to collimate (with a cadmium or boron slit system) a portion of the neutrons which are diffusing out through the bounding surface of 1 For the spectrum of neutrons leaving a pile, see Fig. 54. Sec. 3B] Sources and Detectors 401 10,000 1000 0.001 20 71 cm->j from face of tank 100 120 140 160 180 200 220 2 Cm from face 260 280 -End of graphite Fig. 46. Activities of bare and cadmium-covered indium foils in the thermal column of the Argonne heavy water pile. The thermal column is of graphite, 5 ft square and 240 cm long. Its front face is separated from the reactor by 71 cm of graphite. The average thermal neutron flux (nv) can be obtained by multiplying the ordinates corresponding to the bare indium foils by 0.10. The number of neutrons per cubic centimeter passing through the indium resonance energy per second (slowing-down density) can be obtained by multiplying the ordinates corresponding to the cadmium- covered indium foils by 0.0019. Measurements due to Seren (unpublished). 402 The Neutron [Pt. VII TABLE 27 Some Available Thermal Neutbon Fluxes (a) Laboratory Sources Strength Thermal nv Source (n/sec) Position (n/cm 2 • sec) 1 curie of Ra-a-Be in 13.2 X 10« 4 cm from the source 8.5 X 10 4 infinite H 2 10 cm from the source 2.9 X 10 4 20 cm from the source 0.31 X 10 4 50 cm from the source 0.0025 X 10 4 1 curie of Ra-a-Be in 14 X 10 6 at source 0.7 X 10 4 graphite column, 10 cm from the source, 0.65 X 10 4 4 ft x 4 ft x 7 ft; on the axis source 2 ft from 50 cm from the source, 0.25 X 10 4 base, on axis on the axis 100 cm from the source, 0.04 X 10 4 on the axis 2 curies of Ra-y-Be 3.2 X 10 6 at source 0.3 X 10 4 (608 grams Be) in 10 cm from the source, 0.2 X 10 4 graphite column, on the axis as above 50 cm from the source, on the axis 0.07 X 10 4 100 cm from the source, 0.008 X 10 4 on the axis (b) The Center of a Nuclear Reactor ' Pile Type Location Power Level (kw) Flux (n/cm 2 • sec) Heterogeneous Graphite — U Oak Ridge, Tenn., U.S.A. 2 Harwell, England (GLEEP) 3 (BEPO) 4 ~10 3 4X 10 3 2 X 10 12 10 8 10 12 Brookhaven, L.I., U.S.A. 5 2.8 X 10 4 4 X 10 12 Homogeneous H 2 0— U 236 Los Alamos, N. Mex., U.S.A. (HYPO) « 6 3 X 10 u Heterogeneous D 2 0— IT Argonne, 111., U.S.A. (CP3) ' Chalk River, Canada (NRX) ' 300 >10 4 10 12 5.8 X 10 13 Chatillon, France (Zoe) 8 —10 3 X 10 10 Oslo, Norway 9 ~100 3 X 10 u Sec. 3B] Sources and Detectors 403 TABLE 27 (Continued) Some Available Thermal Neutron Fluxes (c) Pile Thermal Columns Flux Pile Position (n/cm 2 • sec) Argonne (CP3) 10 inner face 3 X 10 s Chalk River (NRX) " 1 meter from inner face 10 9 Chatillon (Zoe) 8 outer face 2 X 10 4 (d) Holes in the Shield Pile Flux (n/cm 2 ■ sec) Oak Ridge 10 10 7 Chalk River (NRX) " 4X 10 7 Chatillon (Zoe) 8 10 6 1 D. J. Hughes, Nucleonics, 6 [2], 5(1950); H. S. Isbin, Nucleonics, 10 [3], 10 (1952). 2 A. H. Snell, Nucleonics, 8 [3], 3 (1951). 3 F. C. W. Colmer and D. J. Littler, Nucleonics, 8 [1], 3 (1951). * Nucleonics, 8 [6], 36 (1951). 6 Physics Today, Jan. 1951, p. 6. 6 Nucleonics, 7 [6], 2 (1950). 7 Nucleonics, 6 [3], 77 (1950); F. W. Gilbert, Nucleonics, 10 [1], 6 (1952). 8 L. Kowarski, Helv. Phys. Acta, 23, Supp. 3, 70 (1950). 9 Bulletin of the Atomic Scientists, 7, 380 (1951). 10 Preliminary reports (unpublished) for NRC Subcommittee on Neutron Standards, by Bernstein and others. 11 E. Almquist, Can. J. Research, A28, 433 (1950). 12 D. G. Hurst, A. J. Pressesky, and P. R. Tunnicliffe, Rev. Sci. Instr., 21, 705 (1950). the moderating medium. However, the beam intensities, which can be attained by this means (especially with easily available sources), are very low, since relatively large masses of moderating material are re- quired to thermalize the source neutrons, and the neutron densities, at the surface of the moderator, are consequently low; in addition, the emerging neutrons have a relatively broad angular spread, so that most of them are absorbed by the collimating slits. The common method of attaining strong thermal neutron beams is to use a "howitzer," which is a relatively large mass of moderator (usu- ally paraffin) with a hole reaching down to a region of high thermal neu- tron flux, close to the source. The hole is usually lined with cadmium 404 The Neutron [Pt. VII or boron or both, thereby providing the necessary collimation for the beam. By proper design, it is thus possible to obtain relatively strong, well-collimated thermal neutron beams, even when only radioactive (a,n) sources are available (J 13). C. Neutron Detectors 1. Induced Radioactivity. Neutrons are capable of inducing a variety of nuclear reactions, many of which result in radioactive product nuclei. Thus, neutrons can frequently be detected by the resulting radioactivity of the exposed substance. Radioactivity is detectable in many ways: Geiger counters for beta- rays of energy sufficient to penetrate the counter walls or, with lesser efficiency, for gamma-rays; electrometers or ionization chambers for alpha-, beta-, or gamma-radiation; scintillation counters, electron multi- pliers, etc. The feasibility of detecting the radioactivity resulting from neutron exposure depends on the lifetime of the induced radioactivity; it cannot be appreciably shorter than the time which must (for experi- mental reasons) elapse between exposure to neutrons and measurement of the resulting radioactivity; on the other hand, the lifetime must not be so long that the radioactive decay rate is negligible. These con- siderations normally have the effect of limiting the lifetimes of possible radioactivities to between ~10 — x sec and ~40 4 years. Consider a detector which absorbs neutrons at the rate of Rq per second, and whose consequent radioactivity decays with a mean life of r sec. The number of radioactive nuclei, N*, is governed by the differ- ential equation dN* N* — = R (100) dt r The activity of the detector (disintegrations per second), t sec after a (constant) exposure of duration t e , is R(t,t e ) = 7?o(l - e~ l ' /T )e- t/T (100a) The observed activity (usually counting rate), R'(t,Q = &R(t,t e ) (101) contains the factor S, the efficiency of the detecting system; S depends on the properties of the radiations involved, on the geometry of the absorber and detector, and on the efficiency of the radiation detector. The activity which would be observed immediately after an infinite irradiation, R'(t,t e ) R ' = #'(£ = 0, t e = oo) = __ _ = S R Q (ioia) (1 — e- ( «/'-)e _ '/ r Sec. 3C] Sources and Detectors 405 is usually referred to as the "saturated activity." Its value is inde- pendent of the schedule of neutron exposure and radioactivity measure- ment. Ro' is usually measured by integrating the activity over a finite energy interval, say from t = t x to t - t 2 . The integrated activity is C(h,ta,Q = f t 'R'&Q dt = i\R'(ti,U) ~ R'(t 2 ,te)] (102) and the saturated activity is p , C(h,k,t e ) Ro — (102a) r(l - e- ( «A)(e- ( iA - e-'sA) The saturated activity is a measure of the neutron flux impinging on the absorber. If S is known, the radioactivity measurements yield R , which can be directly interpreted in terms of the neutron flux <j> = nv. We consider two types of measurement: (1) The absorber is embedded in an isotropic neutron flux whose spectrum is given by <j>(E), the density of neutrons per unit energy interval, at the energy E, multiplied by the neutron velocity correspond- ing to E. If, furthermore, the absorber is weak, 1 the rate of neutron absorption is R = Nvf<j>(E)<r a (E) dE (103) where N is the density of absorbing nuclei and V the volume of absorb- ing material. (NV is the total number of absorbing nuclei.) In general, both cf>(E) and a(E) are relatively complicated functions .of E, and a measurement of R serves only to determine the integral. However, there are some interesting special cases: Suppose that the neutron density has a Maxwell distribution *(») = vM(v) = Av 3 e - (v/v ' )2 (93a) and the cross section follows the 1/v law, <r a (v) = (94a) v Then R Q = NVc a 'v'A f v 2 e~ <»A') 2 dv NVAW)*tr a '7rX 4 1 By which we mean that its presence does not disturb the neutron density, (103a) 406 The Neutron [Pt. VII The total flux is x $ = I M(v) dv = Jo A{v') iM whence 2 (93b) NV$a a 'ir 1A NV$c a ' R = = ( 10 3b) 2 1.128 Thus, for a Maxwell neutron distribution and al/» detector, the mean detector cross section is _ er a V IT 7 * , <f a fa = (95a) ,v 2 1.128 where a- a ' is the absorption cross section corresponding to the neutron energy E' = kT. Suppose, on the other hand, that the absorber has a single sharp resonance at the energy E r . (The effect of thermal neutron absorption can be eliminated by surrounding the detector with cadmium.) The cross section can be represented by a delta-function: c a {E) = P 8(E,E r ) (104) where p = j<r a (E)dE = -<r r (104a) for a resonance, of peak cross section <r and width r, which follows the Breit-Wigner formula. Provided that <t>(E) does not vary appreciably over the resonance, Ro = NV P <t>{E r ) (103c) When, as is frequently the case in the resonance region, 4>{E) = - (96) Hi Eq. (103c) gives NVpB Ro = —=— (103c') E r If more than one resonance is involved, Ro = NVBYi ~ (103c") i Eri The expression E T = J v ~V~ (104b > is called the "resonance integral." Sec. 3C] Sources and Detectors 407 Usually, the insertion of an absorber into the neutron flux causes a perturbation of the flux measured, so that the activity of the detector is not an exact measure of the true flux in the absence of the absorber. This perturbation arises from two causes: a reduction of the neutron density inside the detector which is due to the absorption of neutrons by the outside layers; and a depression of the flux directly outside the detector due to the finite neutron absorption. Such effects have been considered by Bothe (B53), who has derived a correction factor /for the reduction of the specific activity of a sphere of radius R, 1 a - = 1 +- / 2 3RL ~2\^R+~L)~ . for R » \ ir (105a) 1 0.34aK - = 1 + — for R « X tr (105b) J A«r In the above expressions, L is the diffusion length (see Section 4) of the neutrons in the medium outside the detector; \ tr is their transport mean free path; a is the average probability that a neutron will be absorbed in a single traversal of the detector. For an isotropic flux ■ « = 1 _ e -» T {l - pT) + n 2 T 2 Ei{-».T) (105') where ft = N<r a is the absorption coefficient, in cm -1 , of the detector and T is the average absorber thickness. (T = R for a spherical ab- sorber.) Ei{ — x) is the exponential integral. Tittle (T12) has shown experimentally that the above expressions also apply without modification to an absorber in the form of a disk of radius R and thickness T. (2) The second situation of interest involves neutron beams. Con- sider a neutron detector of area (perpendicular to the beam direction) A and thickness T. The absorption rate is Ro = aU(E)(1 - e-" r ) dE (103') An absorber for which fiT = NTc a (E) » 1 is said to be "black" for neutrons of energy E. Since ft (black) = A J 0(2?) dE (103 'a) such an absorber gives a direct measure of the total neutron flux. At 408 The Neutron [Pt. VII the other extreme, a "transparent" detector is one for which fiT <JC 1, in which case R (transparent) = ATN f<t>(E)* a (E) dE (103'b) Owing to the large variation with energy of slow-neutron cross sec- tions, most absorbers are transparent at some energies, black at some, and translucent at others. 2. The Szilard-Chalmers Reaction. In many nuclear reactions, e.g., (n,p), (n,a), (ft,/), the product nuclei differ in their chemical properties from the target nuclei, and can therefore be separated from the target with relative ease. For such detectors the radioactivity from large masses of absorber can be reduced to moderately small samples for counting, and the absolute detection efficiency can be made relatively high. However, many of the important neutron reactions — (n,y), {n,n'), (n,2n) — lead to isotopes of the target element. For such detectors the specific activity of the absorber may be too small to permit accurate measurement of relatively weak neutron fluxes. To overcome these difficulties Szilard and Chalmers (S62) in 1934 devised a technique for separating radioactive nuclei, induced by the (n,y) reaction, from their isotopic environment. This technique takes advantage of the fact that the emission of gamma-radiation after slow- neutron capture causes a recoil of the product nucleus; the recoil energy is usually sufficient to disrupt the molecular bond and, thereby, to change the chemical state of the product nucleus as compared to a nor- mal nucleus in the medium. The radioactive nuclei can then, in favor- able circumstances, be separated chemically from the normal nuclei, with efficiencies as high as 100 percent. For the emission of a single gamma-ray of energy E y (in Mev) from a nucleus of mass number A, the nuclear recoil energy E (in ev) is 536E y 2 , N E = (106) For typical values, E 7 = 7.5 Mev and A = 100, the recoil energy is E = 300 ev, which is considerably in excess of that usually necessary to disrupt chemical bonds, say ~5 ev. Even if two or more gamma-rays are emitted after slow-neutron capture, all but a few percent of the re- coiling nuclei will usually have E > 5 ev. In their original experiment Szilard and Chalmers irradiated ethyl iodide with neutrons from a Ra-a-Be source (most of the absorptions Sec. 3C] Sources and Detectors 409 are due to slow neutrons) and extracted the radioactive I 128 (25-min half-life) by shaking with an aqueous reducing solution containing a trace of free iodine as a carrier. Similar techniques have been applied to other halogens, and methods have been worked out for separating the product nuclei, due to the (n,y) reaction, of many other elements. Specific Szilard-Chalmers processes for different substances are de- scribed and discussed in excellent summaries (with references) of the technique by McKay (M14), by Broda (B68), and by Barnes, Burgus, and Miskel (W2). The Szilard-Chalmers reaction has frequently been used to obtain radioactive samples of high specific activity, especially when only rela- tively weak neutron sources were available. Thus, von Halban, Kowar- ski, and Magat (H6) employed a solution of bromine to measure the intensity of neutrons in the cosmic radiation. The strength of relatively weak fast-neutron sources is frequently measured by stopping the neutrons in a water solution of potassium permanganate, and removing the radioactive Mn 56 (2.6-h half-life) as Mn0 2 by filtration through fine paper or sintered glass (E6) ; the efficiency of this Szilard-Chalmers reaction is greatest for a neutral or slightly acid solution. Broda and Rieder (B67) have shown that the same reaction can be used to detect fast neutrons. 3. Ionization Chambers and Proportional Counters. Many neutron reactions involve the prompt emission of a charged particle. Such reac- tions can be detected, through the ionization caused by the emitted particle, in an ionization chamber or a proportional counter. In addi- tion to whatever other distinctive properties such detectors may possess, they have the advantage over induced radioactivity that they can be used to observe neutrons in situ during the irradiation, and will reflect short-time fluctuations in the neutron density; a radioactive detector, on the other hand, is inherently a time-integrating device. Owing to the relatively short range of the heavy charged particles resulting from neutron reactions, the target nuclei must, in general, be inside the chamber. They can be introduced into the chamber either as a constituent of the gas with which the chamber is filled (and in which the detected ionization takes place) or as a coating on the inside walls. In ionization chambers the ionization of the gas is measured (without amplification in the chamber) either as a current or, in the case of pulsed ionization chambers, as individual pulses associated with single nuclear disintegrations. Proportional counters differ only in that the ionization is amplified in the gas of the chamber. In both cases, the observed charge or pulse height is proportional to the energy dissipated in the 410 The Neutron [Pt. VII chamber (although the constant of proportionality may vary somewhat with the position in the chamber at which the event occurs). The construction and use of ionization chambers and proportional counters has been discussed in Part I of Volume I. The following is intended as a brief summary of the main considerations involved in the use of such devices to measure neutron fluxes: When a neutron-detecting ionization chamber or proportional counter is placed in a neutron flux <j>(E), the observed intensity, / (ionization rate or counting rate), is determined by the same factors as the saturated activity R ' of a radio- active detector (Section 3C1) with the important difference that the de- tection efficiency is, in general, a function of the neutron energy. Thus, we have (neglecting the variation of efficiency with position in the cham- ber) : for a chamber in an isotropic neutron flux, I = NV f&(E)^(E)a(E)f(E) dE (103") where the symbols are as previously defined and f(E) is given by Eqs. (105a), (105b), (105'); for a chamber in a neutron beam, I = A £&(E)<}>(E)(1 - e-* r ) dE (103'") The efficiency function, &(E), depends on the particular reaction used, on the chamber geometry, on the method of introduction of the reacting nuclei, and on the particular detection method used. The following dis- cussion applies only to neutron reactions in which all the energy is car- ried away by charged particles. Reactions involving the emission of neutrons as well as charged particles (of which the most important is neutron scattering, especially by protons) will be taken up in a separate section. We consider a number of the most widely used detection schemes : (la) / = ionization current; the reacting nuclei are in the chamber gas: 8(E) = XX (E + Q) X G{E) (107) where E is the neutron energy and Q is the reaction Q value. (Note : For Q < 0, S = for E < -[(A + 1)/A]Q.) G(E) is a geometric factor which depends on the chamber shape, on the gas pressure, and on the range-energy relationship for the emitted charged particles in the cham- ber gas. It represents the average (over the chamber) fraction of the re- action energy which is dissipated in the chamber gas; G(E) — > 1 as the range of the emitted particles becomes small compared to the chamber dimensions. The calibration constant K can be determined by measur- Sec. 3C] Sources and Detectors 411 ing the ionization produced by a known flux of known energy; alterna- tively, K can be determined by observing the ionization resulting from the insertion into the chamber of a calibrated source of polonium alpha- particles. (lb) / = ionization current; the reacting nuclei are introduced as a coating on the chamber walls. If the coating is thin compared to the range of the emitted charged particles, the situation is similar to that for a gas-filled chamber, except that only ~ \ the reaction energy is dis- sipated in the chamber gas. As the coating thickness is increased, the emerging charged particles will lose energy in traversing the coating. For a thick coating (T > range of the emitted charged particles) only those reactions taking place within a layer of thickness equal to the range of the emitted charged particles can be effective in producing ionization in the chamber gas. The efficiency function of an ionization chamber with a given coating can, in general, be computed. In addition to its dependence on the specific properties of the reaction, the effi- ciency may also depend on the angular distribution of the neutron flux being detected. The form of &(E) is quite different for the counting of individual pulses. In general, the associated electronic equipment is designed so that only pulses of magnitude greater than a predetermined value are counted. (This type of discrimination makes it possible to use pulse counters even when the neutron flux is accompanied by a considerable background of gamma-radiation.) A pulse detector in which all pulses greater than a given size are counted is called an "integral" detector. It is also possible to arrange the electronics so that only those pulses are counted whose sizes lie between two definite (usually variable) limits; such arrangements are called "differential" detectors. (2) I = counting rate; integral detection. If the reacting nuclei are in the chamber gas, &(E) is, as a first approximation, given by a step function, having the value zero for neutron energies less than E t , and one for E > E t ; E t is the neutron energy for which the ionization of the reaction products is just sufficient to produce a pulse of the required size. However, in a chamber of finite size the step function must be modified by a geometric factor G(E), to take into account the possibility that some products of an otherwise detectable reaction may dissipate part of their energy in the chamber walls and give rise to pulses smaller than the detectable minimum. When the target nuclei are coated on the chamber walls the same considerations apply as in scheme (lb). (3) / = counting rate; differential detection, with the reactions occur- ring in the gas of the chamber. Ideally (for a chamber of infinite dimen- 412 The Neutron [Pt. VII sions) &(E) is given by the difference between two step functions with thresholds E t \ and E t 2- In practice, the effect of the finite size of the chamber is to make S smaller than 1 between these limits and to add to &(E) a tail extending beyond E t2 . When the reacting nuclei are introduced as a wall coating, differential detection is useful only if the coating thickness is small compared to the range of the reaction products. (a) Detection by the B 10 (n,a) Reaction. Because of its large cross sec- tion, the B 10 (w,a) reaction is extensively used in slow-neutron detectors. The reaction is highly exoergic, Q = 2.78 Mev. However, most slow- neutron captures lead to the 0.48-Mev excited state of Li 7 (which decays to the ground state by gamma-ray emission); the ionization energy re- leased in such captures is 2.30 Mev. Bichsel, Halg, Huber, and Stebler (B29) and Petree, Johnson, and Miller (P14) have measured the ratio of captures leading to the ground and to the 0.48-Mev excited states for neutron energies from thermal to 3.9 Mev. The ratio follows a smooth curve, from ~0.07 for thermal neutrons to a maximum of somewhat greater than 2 at 1.9 Mev, and then falls to ~1 above 2.5 Mev. The (n,a) cross section of normal boron element follows the 1/v law up to energies of at least 1 kev, with <r'(y' = 2.2 X 10 5 cm/sec) = 710 barns. Its behavior in the intermediate range is not so well established, but its general features have been determined; there is a resonance at 1.9 Mev (G20, A2, P14). The isotope B 10 has a natural abundance of 18.83 percent. Its iso- topic (n,a) cross section is greater than that for natural boron by the factor 5.31 ; i.e., </(B 10 ) = 3770 barns. 1 The availability of B 10 enriched boron thus makes possible the construction of neutron detectors with high efficiency over a wide range of neutron energies. Boron-containing ionization chambers and proportional counters have been operated under a variety of conditions, as current-measuring de- vices and disintegration counters, with the boron introduced as a gas or as a wall coating. Because of the large Q value, 8(E) is essentially energy independent in the slow- and intermediate-neutron energy ranges. Furthermore, the large Q value makes it easy to discriminate between pulses resulting from neutron capture and background pulses arising from relatively strong gamma-ray intensities; hence, boron- containing chambers are normally operated as integral pulse counters. BF 3 is usually used as the chamber-filling gas. It has relatively good ionization chamber and proportional counter characteristics, provided that the gas is free of impurities. Techniques for the construction, fill— 1 See the footnote to Table 11, p. 339. Sec. 3C] Sources and Detectors 413 ing, and operation of BF 3 counters have been discussed by a number of authors (S15, B31, F39, B56). For a chamber containing BF 3 at one atmosphere pressure, the slow- neutron absorption coefficient is M (normal BF 3 ) = 0.0191 (0.025/E)' A cm -1 (i (B 10 F 3 ) = 0.101 (0.025/^)* cm -1 (E is in ev.) Thus, a typical counter (of average thickness ~25 cm) would vary in effectiveness from black to transparent over the ther- mal neutron range. However, most BF 3 counters — even B 10 F 3 — are transparent for epithermal neutrons. For a slow-neutron-induced dis- integration in such a chamber, the range of (93% of) the alpha-particles is -~0.8 cm. (The Li 7 fragment, which carries off f{ i of the energy, has a much smaller range.) Thus, for chambers of reasonable dimensions, the wall effect correction G(E) is relatively small, and &(E) « 1. Chambers in which the boron is introduced as a wall coating are less extensively used for slow-neutron detection. The thickness of the coat- ing must be kept very small, both because of the small range of the reac- tion products and because of the strong neutron absorption of boron (if the neutrons must penetrate the coating from the outside). Hence, such chambers are usually quite transparent to neutrons. It is possible to decrease the transparency, without decreasing the efficiency, by design- ing the chamber to contain a multiplicity of thin boron layers (L24). (b) The Long Counter. The sensitivity of BF 3 counters decreases rapidly with neutron energy, becoming prohibitively small, for most counters, in the intermediate-energy range. In order to increase the sensitivity to intermediate and fast neutrons, experimenters have fre- quently immersed the counter in a moderating medium, usually water or paraffin. However, because of the strong dependence of the sensi- tivity of such arrangements on the source energy and on the geometry, the results of this method of neutron flux measurement are usually very difficult to interpret, especially when the source has a heterogeneous energy distribution. Hanson and McKibben (H32) have devised an arrangement which is uniformly sensitive to neutrons from ~10 kev to ~3 Mev. This ar- rangement, the "long counter," consists of a cylindrical BF 3 counter (diameter ~J in., length ~10 in.) placed along the axis of a paraffin cylinder ~8 in. in diameter. Two of their successful long counter de- signs are shown in Fig. 47. The long counter is used in observations on neutron beams which enter the circular face (right-hand side in Fig. 47a) of the arrangement preamplifier Aluminum tube Long Counter, 8"0.D. Case of 0.050" sheet iron. Removable aluminum cylinder containing BF, tube High voltage -3'^) J* Shielded Long Counter (a) 2 3 4 Neutron energy (Mev) (b) Fig. 47. (a) Two long counters, designed by Hanson and McKibben (H32), with uniform sensitivity (to within ~10 percent) from 10 kev to 3 Mev. The second, or shielded, counter is designed to operate in a region of relatively high scattered neu- tron background, (b) Sensitivity vs. neutron energy of the shielded long counter with and without holes in the front face of the paraffin: 414 Sec. 3C] Sources and Detectors 415 and diffuse parallel to the axis. The second arrangement shown is de- signed for situations in which there is an appreciable background of scattered neutrons impinging on the paraffin from all directions. The uniform energy sensitivity is based on the approximate cancellation of two effects: (1) low-energy neutrons do not penetrate far into the paraf- fin, and have an appreciable probability of being turned around and escaping out of the front face; (2) high-energy neutrons penetrate much farther before being slowed down, but have an appreciable probability of escaping through the sides. The success of the arrangements of Han- son and McKibben is based on a proper choice of the diameter of the paraffin cylinder. In the second design, holes can be bored into the front face to decrease the loss of low-energy neutrons (Fig. 47b). For the shielded counter shown in Fig. 47a, the sensitivity does not vary by more than ~10 percent between 10 kev and 3 Mev, as shown in Fig. 47b. At lower energies the sensitivity decreases to -~70 percent for thermal neutrons; at higher energies the sensitivity again falls off (it is ~85-95 percent for Ra-a-Be neutrons). The absolute sensitivity of the counters is such that they give ~1 count per 10 s neutrons emitted isotropically from a source 1 meter from the face. (c) Fission Chambers. Fission of the heaviest nuclei by neutrons provides an ideal source of ionizing particles for pulse detectors. Since the fission fragment energies are -~100 Mev and their ranges are very small, the pulse due to a fission fragment is easy to distinguish from the background due to other ionizing radiation (e.g., pulses due to alpha- particles, spontaneously emitted from most fissionable nuclei). Thus, fission chambers are almost always operated as integral pulse-counting ionization chambers. The fissionable nuclei may be introduced into the chamber as a gas (e.g., UF 6 ) or as a wall coating. Owing to the general unavailability of gases containing the heaviest elements or the inconvenience in handling these gases, or both, the latter method is usually used; uranium, for ex- ample, is easily obtainable in a solid oxide or nitrate form. In either event, the efficiency function, &(E), is essentially independent of neutron energy, since the Q value of the (n,f) reaction is so great. (We neglect the possibility that the coating may be so thick as to prevent the neu- trons from penetrating into the chamber.) Fission chambers containing thermally fissionable nuclei (U 233 , TJ 235 , Pu 239 ) are efficient thermal neutron detectors. Chambers containing natural uranium, or uranium enriched in the U 238 isotope, can be used as fast-neutron detectors, with an effective threshold of ~1.5 Mev; thorium, protactinium, and neptunium can also be used in the same fashion (see Section 2C4). 416 The Neutron [Pt. VII Bismuth undergoes fission by ultrafast neutrons; it has a threshold of ~50 Mev and a cross section, for ~84 Mev neutrons, of ~0.05 barn (K7, W26). Ionization chambers coated with bismuth have been used as specific ultrafast-neutron detectors (W26). (d) Proton Recoil Detectors. Perhaps the most common method of detecting and measuring fast-neutron fluxes involves the observation of proton recoils from neutron-proton scattering. Use of this reaction has the advantage that the cross section is comparatively large (14 barns at E = 0.1 Mev; 4 barns at E = 1 Mev; 1 barn at E = 10 Mev; see Fig. 2). The main disadvantage of the proton recoil method is that for a given incident neutron energy, E n , the proton recoil energies, E, are uniformly distributed between and E n ; the recoil energy as a function of the angle 6 in the laboratory coordinate system (0 is confined to the interval 0-90°) is E = E n cos 2 6 (108) Let us assume a monoenergetic neutron flux <j>(E n ) incident on a trans- parent chamber containing NV atoms of hydrogen. Neglecting, for the moment, the effects of the finite dimensions of the chamber, the number of pulses corresponding to the proton energy E is NV<j>(E n )<r(E n ) cLE P(E n ,E) dE = VK I ~ (109) E n for E <E n ; P(E n ,E) = for E > E n . The chamber may be operated as an integral counter, in which all pulses are counted whose energy exceeds the bias energy E t . In this case, the counting rate is E " IV(tf n ) I(E n ,E t ) = f P(E n ,E) dE = NV4>(E n ) JBt (E n - E t ) (110) Proton pulses of energy less than ~0.1 Mev are usually too small to be distinguished from the background. In the region 0.1-10 Mev, the neutron-proton scattering cross section follows, to a good approximation, a l/v law; i.e., <r(E n ) ^ (E t /E n ) y2 <r(E t ). In this energy range I(E n ,E t )^NV<j>(E n )a{E t ) (—J = NV<j>(E n )<r(E E n l~E t ~ l «Kt) (110a) The sensitivity of such an integral pulse counter is plotted as the solid curve in Fig. 48. S(E n /E t ) has the property, in this energy range, of Sec. 3C] Sources and Detectors 417 being relatively independent of the neutron energy; its value is within 30 percent of the maximum for 1.52^ < E n < U.5E t . For neutrons of E n > 10 Mev, the neutron-proton cross section can be approximated by <r(E n ) = {E t /E n )c{E t ), whence «®-®'[®-' (110a') This sensitivity function is plotted as the broken curve in Fig. 48. In 1.0 f I 0.! I 0.6 8 0.4 7 / 1 i f\ II -if \ \ 1/ 41 H S \ 1 > If (J v X 1 •». 1 li ^». -- ->■ \\ -— -- — ~-r=r 0.2 ! 10 16 18 Tig. 48. The sensitivity function S(E n /E t ) of an integral pulse-counting hydrogen- filled chamber (neglecting wall effects) vs. x = E n /E t . The solid curve is for <r(E n ) = (Ei/E n ) ' A cr(Et), a good approximation to the neutron-proton scattering cross section for the range 0.1 Mev < E n < 10 Mev. The broken curve is for a{E n ) = (E t /En)<r(Et), which applies for E n > 10 Mev. The curves are normalized to a value of 1 at the maximum. this energy range the integral counting rate is much more strongly de- pendent on E„, However, for neutron energies above ~20 Mev, the angular distribution in neutron-proton scattering is no longer spherically symmetrical in the center-of-mass coordinate system, and the formulas developed above must be correspondingly modified. Hydrogen-filled chambers are frequently operated as ionization cham- bers. For such operation the chamber sensitivity to neutrons of energy E n is I'{E n ) = I 1 J n P(E n ,E)E dE = NV4>{E n )*(E n )E n (111) for neutrons of energy up to ~20 Mev, and neglecting wall effects. How- ever, ionization chambers are particularly sensitive to backgrounds of all kinds. Since most neutron fluxes are accompanied by an appreciable 418 The Neutron [Pt. VII gamma-ray intensity, the background presents a serious problem. One device for eliminating such background effects is to employ two chambers of identical geometry, one filled with the hydrogen-containing gas and the second filled with argon. The external electronic amplifications are adjusted in a pure gamma-ray beam so that both chambers give the same current. The difference in the currents, when the chambers are placed in a neutron flux, can be attributed to the proton recoils from neutron scattering. The effect of the finite dimensions of the chamber is to decrease the sensitivity by a factor which depends on the neutron energy and which can be computed for most chambers. The geometric factor, G(E n ), is a monotonically decreasing function of E n . Unfortunately, the recoil proton ranges are in general comparable to the chamber dimensions, so that G(E n ) usually represents a sizable correction. For a given cham- ber geometry the correction becomes less important as the gas pressure is increased. Thus, the development of techniques for the operation of ionization chambers and counters at high pressure (W35) is of consider- able importance for the future of fast-neutron spectroscopy. Alterna- tively, since organic phosphors (solid and liquid) have a high hydrogen content, it is possible to employ such materials for fast-neutron counting by observing the scintillations resulting from the recoil protons (J3, 06) ; x the recoil proton range is usually small compared to the dimen- sions of the phosphor. Unfortunately, the pulses from such counters do not appear to be strictly proportional to the recoil proton energy. The above discussion applies to monoenergetic neutron sources. If the neutron source is heteroenergetic, the pulse height distribution is a superposition of pulses due to all the neutrons : r°° r x <t>(E n )a(E n ) dE n P(E) dE = I P{E n ,E) dE n dE = NV dE I — — — — (109') Je J e E n The function <j>(E n )a(E n )/E n can be obtained by differentiation of the curve of P(E) vs. E. However, in order to determine <t>(E n ) vs. E n to a reasonable accuracy, the curve of P(E) vs. E must be determined to a much greater — frequently unattainable — accuracy. The use of integral counting to obtain 4>{E n ) vs.'E n is even less satisfactory in that it re- quires exceptionally high precision. When the incident neutrons are in a beam (unique direction of motion) it is possible to choose, for counting, only those proton recoils whose directions are the same as that of the incident neutrons. In such a de- 1 An ingenious application of this principle has been devised by W. F. Hornyak, Rev. Sci. Instr., 23, 264 (1952). Sec. 3C] Sources and Detectors 419 vice it is necessary to collimate the protons. However, the collimation need not be too sharp; a proton recoil at, say, 20° from the incident neutron direction has an energy E = 0.88^„ (Eq. 108). Assuming a collimating system which accepts protons within the relatively small angle 6 , and neglecting the small variation of pulse heights within the cone of acceptance, P(E) = NV<KE)*(E)f(fi ) (109") where AE 6 2 f(!h) = — = 1 - cos 2 O = — (108a) (d may be a function of the proton recoil energy E.) The smaller the angle do, the smaller is the value of f(ff), but the sharper the energy reso- lution of the device. Thus, in common with almost all problems in spectroscopy — neutrons or otherwise — the practically attainable reso- lution is primarily determined by the strength of the source. Many arrangements can be conceived for collimating the observed protons. One possibility is to detect the protons in a long cylindrical chamber, the incident neutrons traveling along the chamber axis, and to divide the chamber into two or more independent regions, requiring coincidences between adjoining regions. In some cases, the last section is operated in anti-coincidence with the preceding sections, thereby de- fining the range of the detected protons. Another possibility is to separate the sections by barriers, with holes in them for the proton collimation. The hydrogen may be introduced into the chamber as a gas or, more usually, as a thin radiator at the incident face of the cylinder. A thin radiator is, by definition, one in which a proton, originating in the back of the radiator, loses only a small fraction of its energy before emerging into the counter. The figures in Table 28 give the approxi- TABLE 28 Some Typical Proton Recoil Ranges Range Range E p Range (mg/cm 2 (mm Mev) (cm std. air) paraffin) paraffin) 1 2.3 3.2 0.036 2 7.1 9.7 0.11 5 34 47 0.52 10 115 160 1.8 15 238 330 3.7 mate ranges of fast protons in standard air and in paraffin, a typical radiator material. It may be seen that, for neutrons of energy ~1 Mev 420 The Neutron [Pt. VII or less, "thin" radiators must be very thin indeed. As the radiator thick- ness becomes comparable to the proton range, the energy resolution of the detector rapidly deteriorates. For thick radiators (thickness > maximum proton recoil range) the situation again arises wherein a monoenergetic neutron produces pulses of all energies up to its own. However, the pulse height distribution is much more complicated than that from neutrons scattered in the chamber gas. For neutrons of energy E n incident normally on a thick radiator, with the chamber counting all pulses above the bias energy E t , the sensitivity function is approximately (B9) S(E n ,E t ) g* ke(E n )E n - y \E n y> - E t %A f (110a") in which the range-energy relationship has been approximated by B(E P ) oc E P 3A . This function has zero slope at E n = E t , and is a mono- tonically increasing function of E n ; hence, such a chamber is most sensi- tive to the highest-energy neutrons in the beam. (e) Other Possibilities. In principle, any neutron reaction can be used as the basis for a neutron detector. Thus, the radiative capture process can be observed by detection of the prompt capture gamma-rays; in- elastic scattering can be detected by observation of the accompanying gamma-radiation. With the recent development of scintillation counter techniques, the efficiency of gamma-ray detection is comparable with that for the detection of charged particles. One application of these techniques to neutron counting is of special interest in that it can be used to provide an efficient detector of inter- mediate-energy neutrons (D16). This technique takes advantage of the fact that, for slow and intermediate neutrons, the B 10 (w,a) reaction goes mainly to the 0.48-Mev excited state of Li 7 , from whose decay the gamma-ray can be detected by a scintillation counter. Since, in such a detector, solid boron can be used, the size being limited only by the penetrability of the 0.48-Mev gamma-ray, high efficiencies are easily obtainable. As previously pointed out, charged particle reactions are easily ob- servable if they take place in a scintillating material. Thus, Hofstadter et al. have detected thermal neutrons, with high efficiency, by the scin- tillations produced, in a Lil(Tl) crystal, as a result of the Li 6 (n,a) reaction (H64). Such reactions can also be observed by allowing the charged particle to impinge, from the outside, on a scintillator. The zinc sulfide screen, so important in the pioneer work on natural radio- activity, has been used in this connection (M38). Other charged particle reactions, both exoergic and endoergic, can be used for ionization chamber and proportional counter materials. Among Sec. 3C] Sources and Detectors 421 these, the N 14 (n,p) reaction, Q = 0.626 Mev, has possibilities as an intermediate-neutron detector, since its relatively low Q value permits identification of the energy of the neutron responsible for the reaction through a pulse height measurement (F7). Unfortunately, the presence of resonances in this reaction, for neutron energies above ~500 kev, limits its usefulness to intermediate-energy-neutron spectroscopy. Other reactions with similar characteristics are He 3 (n,p) and Cl 3S (n,p). Finally, recoils from elastic scattering by light nuclei other than H 1 can be used for neutron detection. In this regard, He 4 recoils have the advantage of shorter range (thereby decreasing the wall effect correc- tion) and less energy spread for a given incident neutron energy. 4. Photographic Emulsion and Cloud Chamber Detectors. Photo- graphic emulsions and cloud chambers have been among the most use- ful tools for the observation of nuclear reactions. Both of these devices have the property that the path traversed by a charged particle appears as a visible track. From the length of the track (range) and the density of ionization (grain or droplet density), it is possible to determine the energy and velocity, respectively, of the ionizing particle. A curve of ionization density vs. residual range for a track uniquely determines the energy, mass, and magnitude of charge of the particle. Measurement of the multiple Coulomb scattering in the photographic emulsion or in the gas or plates in a cloud chamber provides a measure of the particle energy. The curvature of the track in a known magnetic field yields the particle's momentum, as well as the sign of its charge. Since track curvature results both from small-angle Coulomb scattering and from applied magnetic fields, the use of magnetic fields is usually confined to cloud chambers, or to regions between two photographic emulsions, since the Coulomb scattering is smaller and the range is greater in a gas than in the emulsion; however, for sufficiently strong fields and particles of high energy, curvature due to magnetic fields has been observed on tracks in emulsions (D14). In essence, photographic emulsions and cloud chambers yield the same sort of information; indeed, the photographic emulsion can be said to be identical with a cloud chamber operated at very high pressure. How- ever, owing to the differences in their mode of operation, emulsions and cloud chambers should be regarded as complementary instruments, the choice between the two being primarily determined by the nature of the problem under investigation. Cloud chambers have the advantage that it is easier to choose and change the filling gas. On the other hand, emulsions are continuously sensitive, and are therefore much more use- ful in experiments involving weak sources. Furthermore, the sensitivity of emulsions is variable over a very wide range — emulsions are available 422 The Neutron [Pt. VII which are insensitive to tracks of ionization density less than that of a fission fragment; at the other extreme, it is possible to obtain emulsions which show tracks of electrons at the minimum of the ionization rate curve. Another advantage of emulsions is their small size and light weight. The main disadvantage in the use of photographic emulsions arises from the short range, in them, of charged particles. (A 1-Mev proton has a range of ~15 /*, a 1-Mev alpha-particle of ~3.5 /j..) As a result, it is necessary to detect and measure tracks with a microscope, and the accumulation of large amounts of data becomes a relatively tedious affair compared to experiments involving electronic detection. How- ever, the wide flexibility and the possibility of unambiguous identifica- tion of reactions frequently more than compensate for this disadvantage. The earliest experiments with photographic emulsion detection of charged particles employed x-ray and "halftone" plates. Thus, Taylor and Goldhaber (T4), as early as 1935, used boron-impregnated Ilford R emulsions, irradiated by slow neutrons, to prove that the B(n,a) reac- tion involves the emission of a single alpha-particle. However, the emulsions available at that time could only detect low-energy charged particles; their lack of sensitivity and uniformity greatly discouraged their use in nuclear physics. Since World War II, tremendous progress has been made in the development of more sensitive emulsions and of techniques for their use, in large measure through the instigation and inspiration of Occhialini, Powell, and their collaborators. Emulsions called "nuclear emulsions," with various degrees of sensitivity, and with various thicknesses, from 25 to 600 fi, are now available. (Thicker emul- sions can be obtained by special arrangement with the manufacturers.) The use of nuclear emulsions has been extensively described by Powell and Occhialini (P25), by Yagoda (Y2), and in a recent summary by Rotblat (R22). Rotblat also gives the most recent data on the composi- tion and properties of the various available emulsions (his Tables I, II, and III) and on processing techniques. There is an extensive literature on cloud chambers. A recent mono- graph by Wilson (W34) summarizes the techniques of cloud chamber operation. Further details on these subjects are given in Volume I, Part I. (a) Detection of Slow Neutrons. Nuclear emulsions contain an appre- ciable amount of nitrogen (0.073, 0.080, 0.11 g/cm 3 , respectively, in Il- ford, Kodak, and Eastman Kodak emulsions). The N 14 (n,p) reaction has a Q value of 0.626 Mev, with a cross section a' = 1.76 barns for thermal neutrons. The protons resulting from thermal neutron capture Sec. 3C] Sources and Detectors 423 (range ~7 n) can be observed and counted in the emulsion, thereby providing a measure of the thermal neutron flux (C33). The sensitivity of nuclear emulsions to slow neutrons can be greatly enhanced by adding small amounts (plates are available with ~1% by weight) of lithium or boron to the emulsions. The (n,a) reactions in such loaded emulsions can be used for measuring even small slow-neutron fluxes with relatively high accuracy (S25, T10, B35, R22). It is also possible to impregnate emulsions with uranium acetate or citrate, and to observe tracks due to fission by slow neutrons (M43). Owing to the high background of alpha-particle tracks from the uranium, such plates must ordinarily be exposed and processed within a short time after preparation. However, emulsions are available whose sensi- tivity to alpha-particles has been greatly decreased, so that only the fission fragment tracks are observable (Ilford Dl, Eastman Kodak NTC). It is also possible to measure slow-neutron fluxes by observing the blackening of x-ray and electron sensitive emulsions resulting from slow- neutron-induced radioactivity (B35). The reactions mentioned above can also be observed in cloud cham- bers, although it is considerably more difficult to obtain a suitable gas containing lithium, boron, or uranium; instead, these elements are usually introduced into the chamber as thin coatings on plates. (b) The Observation of Proton Recoils. Nuclear emulsions contain an appreciable hydrogen content (~0.04-0.05 g/cm 3 in dry emulsions; considerably more in a moist atmosphere). Thus, irradiation with fast neutrons will give rise to proton recoil tracks. Because of the variation of the proton recoil energy with angle, accord- ing to Eq. (108), measurement of the energy of a proton recoil will yield the neutron energy only if the relative directions of neutron and proton are known. In other words, to use proton recoils for neutron energy and flux measurements, it is necessary for the incident neutrons to have a well-defined direction. (The previous discussion of proton recoil ionization chambers and proportional counters is fully applicable to this section.) Thus, in measuring a fast-neutron flux distribution by ob- serving the proton recoil range spectrum, it is necessary to set up strict criteria for the acceptance of tracks. Such criteria usually involve the choice of a limiting angle, O of Eqs. (109") and (108a). In observing tracks which "dip" in the emulsion, it must be kept in mind that emul- sions shrink by a factor ~2 in being processed. It is relatively easy to make corrections for the background due to other (n,p) reactions in the emulsion — say on N 14 — or due to scattered neutrons which strike the emulsion from all directions. Since the recoil 424 The Neutron [Pt. VII protons from neutrons in the beam are confined to the forward hemi- sphere and the background protons are usually distributed with approxi- mate spherical symmetry, measurement of the proton recoil spectrum in the backward hemisphere can provide the necessary data for this correction. For neutrons of energy greater than ~1Q Mev, an appreciable frac- tion of the recoil protons leave the emulsion before coming to the end of their ranges; the range of a 10-Mev proton is ~600 n of emulsion. Thus, as the neutron energy is increased, an appreciable correction must be applied for the loss of tracks. This correction can be decreased by using 0.3 ^.0.2 Li . \ a a J> ^ s 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 •E„(Mev) Fig. 49. Li 6 (n,a)H s cross section for normal lithium (7.4%Li 6 , 92.6%Li 7 ) due to J. M. Blair et al. (unpublished) (G20, A2). thicker emulsions. For very fast and especially ultrafast neutrons, it is more accurate to measure the proton energy by the track's grain density, or by the small-angle scattering. In the intermediate-energy range, the proton recoil method is limited by the short range of the recoil protons, which renders inaccurate the measurement of range and of angle. Most experimenters regard 1 Mev as a practical lower limit of the neutron energies to which the technique is applicable. Recently, Nereson and Reines (Nl) have shown that, by careful application of the technique, accurate neutron flux measure- ments can be made down to E n = 0.5 Mev, and possibly pushed (with poor resolution and large corrections) to 0.3 Mev. The use of hydrogen-containing cloud chambers for neutron detec- tion is subject to the same considerations as have been outlined above. (c) Other Reactions — Especially Li 6 (n,a)H 3 . For the detection of fast neutrons by a charged particle reaction in nuclear emulsions, it is neces- sary that the cross section for the reaction in the emulsion be at least comparable with that of neutron-proton scattering, and that the reaction products be distinguishable from the recoil protons. In the energy range ~0.1-10 Mev, the Li 6 (n,a) reaction, Q = 4.79 Mev, has the desired Sec. 3C] Sources and Detectors 425 properties. The (n,a) cross section of normal lithium (7.4% Li 6 ) is shown in Fig. 49 for neutron energies up to 0.8 Mev. In order to overcome the large proton recoil background, it is neces- sary to employ emulsions impregnated with enriched Li 6 . Such emul- sions have been used by Keepin and Roberts (K5) (Eastman Kodak ">B*' ,1 ' > ''V- . I I. 7 •■■ \ **■• "' \ •09 % \ t Wmi§m $P$$l)!0&i>'- ||E >V: Fig. 50. Photomicrograph of a Li 6 (n,a)He 3 disintegration in a Li 6 loaded NTA emulsion. The lower track is due to the alpha-particle. The background tracks are mainly due to proton recoils. Courtesy J. H. Roberts (private communication). NTA, with 0.04 g/cm 3 Li 6 loading), who have successfully demonstrated the usefulness of this method of fast-neutron detection and flux measure- ment. In the measurements of Keepin and Roberts, the range of the alpha- particle and of the triton are measured, as well as the angle between them. They have found that the best energy resolution, ±0.1 Mev for neutrons from to a few Mev, is obtained by selecting only those events for which the angle between the particles is between 175° and 180°. It is necessary to use special processing techniques to permit discrimination between the alpha-particle and triton tracks. A photomicrograph of a typical event is shown in Fig. 50. 426 The Neutron [Pt. VII The above-mentioned resolution of ±0.1 Mev has been achieved for collimated neutron sources. With isotropic neutron fluxes, for which this method is possible where the proton recoil method cannot be used, such good energy resolution has not been attained (R12). For neutrons of energy greater than ~5 Mev, the very large back- ground of protons of comparable range renders the use of the Li 6 (w,o:) reaction relatively inaccurate. However, for E n > 10 Mev, the B 10 (n,2a)H 3 reaction has been successfully used for neutron detection with boron-impregnated emulsions (L5). D. Slow-Neutron Spectroscopy Probably more work has been done, and more information obtained, on the properties of neutrons between and 10 ev energy than on all the rest of the neutron energy spectrum. This was certainly true before and during World War II, and continues, to a somewhat lesser extent, to be true today. This emphasis has not been entirely due to the greater ease of producing and detecting slow neutrons. The existence and the properties of distinct, narrow, slow-neutron resonances provide a con- firmation and a test of the ideas of Bohr and his followers on the proper- ties of the compound nucleus, and on the possibility of a statistical treatment of nuclei. Furthermore, the wave properties of the neutron and the fact that slow-neutron wavelengths are of the order of inter- atomic distances in molecules, liquids, and solids hold forth the intriguing prospect of using slow neutrons as a tool for the study of the structure of matter. While a number of such studies were carried out in pre- World War II days, this aspect of slow-neutron physics is only now, owing to the availability of the strong sources from piles, bearing fruit; these studies will be the subject of Section 5. The history of the development of techniques for slow-neutron investi- gations was summarized in Section 2A. In this section these techniques are discussed in detail. Although many types of experiments can be designed to measure the variation of some effect or other with neutron energy, the quantity most frequently measured is the cross section. The cross section is usually measured by a transmission experiment. A "good geometry" transmission experiment measures the total cross section, <r = £) cr,-, the sum of the cross sections for all processes which i remove neutrons from a beam; these include scattering as well as ab- sorption processes. The quantity measured in a transmission experi- ment is the attenuation y = e~ N ° T (112) Sec. 3D] Sources and Detectors 427 the ratio of neutron intensities at the detector with and without an absorber (NT = number of absorber atoms per square centimeter of absorber *) in the beam. Such a measurement does not require any knowledge of the value of the neutron flux. In order to obtain maximum accuracy in a minimum of counting time, especially if the available beam intensities are not very great, it is necessary to make a proper choice of the absorber thickness. The opti- mum thickness, which depends on the details of the experimental arrangement and on the background, is, over a wide range of experi- mental conditions, for NaT « 1-2 (R17, R9). Since scattering is always one of the components of the total cross section, it is necessary, in a good geometry experiment, to correct the measured transmission for those scattering processes in which the de- flected neutron strikes the detector. 2 The smallness of the correction is the measure of the goodness of the geometry. In order to make this correction, it is necessary to know the magnitude and the angular dis- tribution of the scattering cross section. Under certain circumstances, a "poor geometry" transmission experi- ment can yield independent and useful information. Consider a beam of neutrons and an absorber, both large in cross-sectional area compared to the detector. Let the detector be placed directly behind the absorber. The attenuation, I/Iq, due to the absorber is, in this case, e~ N °' T , where a' = <r — ^<x sc (provided that the scattering is symmetrical with respect to 90°) since half of the scattered neutrons leave the absorber through the back face. In all these considerations we assume that the neutrons have a negligible probability of making more than one nuclear encounter in the absorber; otherwise, the problem approaches one of diffusion, to be considered in Section 4. A still more extreme case of poor geometry is that of a point source completely surrounded by a (spherical) absorber. In this case, if the detector is sufficiently far from the source — so that the distribution of scattered neutrons also resembles that of a point source — the meas- ured cross section is a a = <r — <r sc . Thus, the results of good and poor geometry transmission experiments enable the determination of both o- sc and <r = <j — a 8c . Experiments in which the geometry is interme- diate between good and poor are rather more difficult to interpret (S65). 1 If the absorber contains more than one type of nucleus, the exponent of Eq. (112) is replaced by NaT = T £ AW. k 2 With respect to this correction, inelastic scattering must be regarded as an ab- sorption process if the detector is insensitive to the inelastically scattered neutrons, and as a scattering process if the detector is sensitive to them. 428 The Neutron [Pt. VII Transmission experiments are, of course, not the only means of meas- uring neutron cross sections. All the techniques whereby specific reac- tions can be used for measuring neutron fluxes (discussed in Section 3C) can be reversed, and cross sections determined if the neutron flux is known. Mention must also be made of the techniques developed by Dempster and co-workers (L4) and by Inghram, Hess, and Hayden (II, H50, H55) for the detection of (strong) neutron absorption and measurement of cross sections by mass-spectrographic observations of the changes in isotopic abundances resulting from neutron bombard- ment. The reader's attention is especially called to a beautiful photo- graph published by Dempster [Phys. Rev., 71, 829 (1947)] showing the decrease in abundance of the Cd 113 isotope when normal cadmium is subjected to intense neutron irradiation. Finally, a transmission meas- urement can be changed into a direct measurement of <j sc by moving the detector out of the neutron beam; however, such a measurement requires either an exact knowledge of the geometry and the neutron flux, or calibration with a scatterer of known cross section, say carbon (H38, H40). 1. Slow-Neutron Monochromators. Devices for singling out neutrons of a specific (usually variable) energy are known as monochromators or velocity selectors. A number of such devices, based on various princi- ples, have been designed and constructed. They all employ, as the slow-neutron source, neutrons emerging from a mass of slowing-down material (paraffin, heavy water, graphite) whose energy distribution is approximately Maxwellian, with an epithermal tail (1/E flux distribu- tion) extending into the intermediate-energy range. As neutron detec- tor, a boron-containing proportional counter is usually employed. For maximum sensitivity over the widest possible range, the counter is filled with enriched B 10 F 3 . Although the type of monochromator employed depends primarily on the nature of the primary neutron source, there are two basic factors which determine the usefulness of a given device: (1) the energy range over which it can be operated, and (2) the resolution at a given energy, which is a measure of the energy spread at the energy under considera- tion. In order to define the resolution, it is necessary to specify the shape of the resolution function. The resolution function can usually be approximated by an isosceles triangle. Unless otherwise specified, a triangular resolution function will be assumed, the monochromator energy being defined as the energy at the apex, and the "resolution" as the energy spread at the base. The characteristics of some of the best existing monochromators are summarized in Table 29. The figures in the third column for the "time of flight" velocity selectors give the ratio Sec. 3D] Sources and Detectors 429 of the time resolution of the instrument (in microseconds) to the flight path (in meters) ; this ratio sets the basic limitation both on the usable energy range and on the resolution of the velocity selector. The last five columns give the energy resolutions which are normally attained in operation. TABLE 29 Characteristics of Some Slow-Neutron Monochromators Usable Range (ev) Resolution 0xsec/m) Resolution (ev) 0.025 0.1 1 10 100 1,000 Argonne thermal chopper (B65) Argonne fast chopper (S18, S19) Columbia modulated cyclo- tron (R4) Harwell electron accelerator (M26) Crystal spectrometer (S56) 0.002-0.2 0.001- 5,000 0.001-10,000 0.001-20,000 0.02 -100 (50) 0.6 1 0.5 0.005 0.001 0.001 0.001 0.0015 0.05 0.05 0.05 0.05 0.01 0.1 0.07 0.1 0.2 1.7 1.2 1.3 5 40 30 20 1,000 1,000 500 (a) Mechanical Velocity Selectors or Choppers. The first slow-neutron monochromator, built by Dunning, Pegram, Fink, Mitchell, and Segre (D20), was based on the principle used by Fizeau in his classic measure- ments of the velocity of light. In this device two cadmium disks were mounted, one at each end, on a shaft. Both disks had a series of uni- formly spaced radial slits; the disks could be displaced by an arbitrary angle with respect to each other. With the disks so misaligned, thermal neutrons, moving along the direction of the shaft, which passed through the first set of slits were absorbed in the second disk, provided that the shaft remained stationary. However, with the shaft rotating, those neutrons which passed through the slits of the first disk, whose time of flight between disks was equal to the time required for the second disk to move through the angle of misalignment, could also pass through the second slits and be detected. Thus, by varying the velocity of rotation or the angular displacement, it was possible to select neutrons of a pre- determined velocity. Although the resolution of this device was poor (Sl3a), the velocity selector was useful in demonstrating the 1/v nature of a number of thermal neutron cross sections. Improvements and further measurements on this velocity selector were carried out by Fink (F29). Taking advantage of the greater thermal neutron fluxes available from the first pile, Fermi, Marshall, and Marshall (F22) constructed a 430 The Neutron [Pt. VII thermal neutron velocity selector based on the same principle, but with a number of significant modifications. In this device a beam of neutrons, from the thermal column, falls on a cylinder whose axis is perpendicular to the beam direction. The cylinder is made up of alternate, lengthwise, thin layers of aluminum (effectively transparent to thermal neutrons) and cadmium, as shown in Fig. 51. Neutrons can pass through the cylinder only when the layers are parallel, within ±3°, to the beam direction. Thus, when the disk is rotated (at speeds up to 15,000 rpm) about its axis, neutrons are transmitted in short bursts. Aluminum v\^ Steel W Cadmium 1 Multiple sandwich of 0.004" to 0.008" cadmium and aluminum foils Fig. 51. Cross section, perpendicular to the axis of rotation, of the mechanical "chopper" used in the thermal neutron velocity selector of Fermi, Marshall, and Marshall (F22, B65). The neutrons which get through the shutter are detected in a bank of BF3 proportional counters, 1.46 meters away. The neutron time of flight between shutter and detector is determined electronically, by allowing the neutron detector to be sensitive only for short periods, delayed by an arbitrary time interval t with respect to the shutter-open times. The synchronization between the shutter position and the de- tector-on time is achieved by attaching to the cylinder axis a mirror which reflects a beam of light into a photocell when the shutter has a given orientation. By varying the position of the photocell, the delay time t can be chosen at will. Only those neutrons will be recorded whose time of flight between the shutter and the detector is equal to t. If the burst time duration and detector-on times are both equal to At, and both have square shapes, the velocity resolution function is a triangle with apex at the velocity v = L/t and base width Av = 2v At/t = 2v 2 At/L, where L = the length of the flight path. Sec. 3D] Sources and Detectors 431 This velocity selector was subsequently improved by Brill and Lich- tenberger (B65), who used it to measure the cross sections of a number of elements in the thermal neutron region. The numbers given in the first line of Table 29 apply to the improved model. Velocity selectors which employ cadmium in the neutron shutter cannot be used for energies above the cadmium cut-off. This limitation Pig. 52a. The fast chopper of Selove, at the Argonne Laboratory (S18, S19). Pho- tograph shows the stator, rotor, and photocell (box) for timing. does not apply to the "fast chopper" constructed by Selove at the Ar- gonne laboratory (S18, S19), which employs, as a neutron shutter, a long cylinder of steel with thin slits cut into the cylinder parallel to the axis, which is also the direction of the neutron beam. In line with this cylinder, which can be rotated about its axis, there is a stationary cylin- der with identical slits. Neutrons can pass through both sets of slits only when the two sets are aligned; when the slit systems are misaligned, the thickness of steel is sufficient to remove neutrons of all energies from the beam. This device is shown in Figs. 52a and 52b. The neutron detection and timing system is, in principle, the same as that of the thermal chopper. In operation, the flight path L = 10 meters and the burst length A* « 6 n sec. The characteristics of this velocity selector are given in the second line of Table 29. The possibil- ity of using such a relatively high resolution device is dependent upon 432 The Neutron [Pt. VII Slit width (W) Fig. 52b. Details of the rotor construction of the chopper in Fig. 52a. 1.0 0.5 "f!'^> Y\ J ^A { • T\ l?~-c r ■• "\o/^~7 W 182 a i.o 0.5 1.0 0.5 10 20 IX sec/m 30 g/cm 2 w 182 W 183 w 184 w 186 1.58 0.054 0.07 0.021 1 o °„ N^ >*~~«» V-S. o ° V W 186 007 0.006 0.023 X ^\ AH ^ * , f\! •\ L>-< \ 1 ' fv/ \J Normal wolfram 8.04E/cm 2 242 240 (3Q.H)(29.8») (22JJ) j 39 U7-3J) 5 5000 500 1 1 1 i — i — i — i — i — i 1 — i — i — i — r 1000 200 100 50 30 20 15 10 8 6 5 4 Fig. 52c. Transmission curves for separated wolfram (W) isotopes, made with the fast chopper of Fig. 52a. Sec. 3D] Sources and Detectors 441 low-energy side of the resonance never falls below 2000 barns. On the high-energy side, on the other hand, the cross section descends abruptly; above ~4 ev the cross section (5 barns) is due entirely to scattering' there being no additional resonances observed up to ~100 ev. A thick- ness of 1 g/cm 2 (~l mm) of cadmium metal will transmit less than 1 percent of all the incident neutrons of energy below 0.3 ev, and ~M of the neutrons of energy 0.45 ev. Only a negligible portion of the Maxwell distribution (for reasonable neutron temperatures) extends into the epicadmium region. On the other hand, if the absorber has a resonance fairly close to the thermal region, the low-energy tail of the resonance may extend to below the cadmium cut-off; in this case, the absorption of a cadmium-covered de- tector will be less than the true resonance absorption, and it is neces- sary to apply a correction (increase) to the cadmium-covered detector reading before subtracting it from the bare detector value, in order to obtain the true thermal neutron absorption. The magnitude of this correction will depend on the thickness of the cadmium shield. Indium is frequently used for a thermal neutron detector because of the magnitude of its cross section (<J ^ 190 barns), the ease of handling indium metal foils, and the convenient half-life of the resulting radioac- tivity (54 min). The lowest indium resonance is at 1.44 ev (<r ^ 35,000 barns, T s 0.08 ev; see Fig. 7c, page 281). The correction to the activ- ity of cadmium-covered indium foils to obtain the true resonance activa- tion has been investigated experimentally by Kunstadter (K26), who gives the following correction formula: True activity = observed activity X e 0A3ST (115) where T is the thickness of cadmium metal in millimeters. 1 (6) The Pile "Danger Coefficient" Method. A useful and accurate method of measuring thermal neutron absorption cross sections has been reported by Anderson, Fermi, Wattenberg, Weil, and Zinn (A25). This method involves the use of a thermal pile (see Section 4), and depends on the fact that introduction of absorbing material into the pile causes a decrease in the pile reactivity. If the pile is operating at a constant neutron flux level, the insertion of the absorber will result in a gradual decrease in the flux; the effect of the absorber can be compensated by displacing the pile control rods by an amount sufficient to maintain the pile at a constant power level. If the compensating motion of the control rod is calibrated against a series of absorbers of known cross section, the required motion of the 1 Below ~0.5 mm of cadmium, the shield is no longer black to thermal neutrons, and the above considerations break down. 442 The Neutron [Pt. VII control rod for an unknown absorber provides a measure of the un- known absorption cross section. This method was first devised in order to ascertain the harmful effects (with respect to pile reactivity) of various contemplated pile construction materials; hence the term danger co- efficient. One of the characteristics of this method is that it measures the absorption cross section, in contrast to the total (absorption plus scat- tering) and the activation (absorption leading to a measurable radio- activity) cross sections. The pile reactivity is relatively insensitive to the scattering properties of the absorber, even if the absorption cross section is comparatively small. Since the products of the particular absorbing reaction (provided they are not neutrons) are of no concern to the pile, this method provides one of the few means of observing ab- sorptions which lead to a stable product nucleus, or to a product nucleus whose radioactivity is difficult to observe; for example, although activa- tion measurements (S24) seemed, at first, to yield an apparent absorp- tion cross section of 0.02 barn for niobium, which has the single isotope 4iNb 03 , the danger coefficient method gives a value of 1.4 barns (A25). The discrepancy was resolved when a previously unrecognized isomer of Nb 94 was discovered. A modification of the danger coefficient method suggested by Wigner greatly increases its sensitivity. This is the method of the "pile oscilla- tor," in which the absorber is intermittently introduced into the pile, resulting in an oscillation of the flux (power) level of the pile. The magnitude of the resulting pile oscillation is a measure of the absorption cross section of the sample. The effect of scattering by the sample is to introduce an out-of -phase component into the pile oscillation; accord- ingly, the effects of scattering and absorption can be separated experi- mentally, and the absorption of samples for which the absorption cross section is only a small fraction of the scattering cross section (e.g., bis- muth) can be measured. The theory of the pile oscillator has been developed by Weinberg and Schweinler (W12). Extensive pile oscillator measurements of thermal neutron absorption cross sections have been made at the Argonne Laboratory by Harris, Muehlhause, Rasmussen, Schroeder, and Thomas (H42), who used an oscillator constructed by Langsdorf (L3), and at the Oak Ridge National Laboratory by Pomerance (P22, P23) with the oscillator of Hoover, Jordan, Moak, Pardue, Pomerance, Strong, and Wollan (H65). A pile oscillator has also been developed and used with the French heavy water reactor by Raievski and Yvon (R2). In using a thermal pile for a danger coefficient measurement of a thermal neutron absorption cross section, it is necessary to correct for Sec. 3D] Sources and Detectors 443 the absorption of an appreciable epithermal neutron component. This correction can be determined by the cadmium difference method. The danger coefficient method can also be applied to the measurement of epithermal absorption cross sections (cadmium-covered foils) (L3). It can be used with intermediate- and fast-neutron piles. In these cases the absorption is an average over the relatively broad neutron flux dis- tribution in the pile. However, in these applications it is much more difficult to correct for the effects of the scattering properties of the sample. 3. Measurement of the Characteristics of Slow-Neutron Resonances. One of the most important problems of slow-neutron spectroscopy is the precise determination of the constants associated with resonances. In the following discussion, we consider only capture resonances, neglecting scattering and associated interference effects; i.e., we assume r « r r J£> r„. (Similar considerations can be applied, with minor modifications, to scattering resonances.) In this case, the Breit-Wigner formula is most conveniently written (E r /E)% ° m - m-^-wn <n6) A resonance is completely described in terms of three parameters: E r , the resonance energy; o- , the peak cross section; T, the full width at half maximum. Equation (116) is not symmetrical about the energy E r , owing to the factor (Er/E)^. The variation of this factor is important for resonances occurring in or near the thermal region. However, for resonances for which T « E r , the deviation of the factor (E r /E) 1A from 1 is negligible over the region of significant values of v(E) ; this situation prevails for practically all the observed resonances for which E r ^ 1 ev. (a) Monochromator Measurements. In principle, a measurement of u{E) vs. E provides all possible resonance data. In practice, the finite resolution of the monochromator has a profound influence on the ob- served shape of the resonance, and on the possibility of deducing the resonance parameters. In the following considerations of the effects of finite resolution we assume a rectangular resolution function, of width R, mainly for ease of computation. The computations can be performed for triangular, or any other shape, resolution functions by numerical iteration if necessary; furthermore, a resolution function of arbitrary shape can usually be approximated with reasonable accuracy by the superposition of a number of rectangles. Since monoehromators measure <r(E) by transmission experiments, the absorber thickness is important to these considerations. We define (1) 444 The Neutron [Pt. VII a thin absorber as one which is at least translucent over the entire resonance, i.e., NT<r < 3; (2) a thick absorber is black in the region of the resonance, NTcr > 10. It should be remarked that the choice be- tween using a thin or a thick absorber is not entirely a question of the available techniques of absorber preparation. The monochromator resolution is the determining factor: for R » T it is impossible to obtain sufficient absorption, and consequently to make any measurement at all, by using a thin absorber. On the other hand, for R < r the use of thin absorbers is, as we shall see, possible and desirable. The available monochromator resolutions have been summarized in Table 29. Since resonance widths are, for heavy nuclei, ~0.1 ev, good resolution (R < T) is possible only below ~1 ev; in the region 1-10 ev, the available resolutions are fair; above ~10 ev, resolutions are poor. The situation with respect to scattering resonances in medium nuclei is somewhat better; however, very few such resonances fall in the slow- neutron region. The considerations of this section can also be applied to measurements in the intermediate and fast regions, where many scat- tering resonances have been observed. (1) Thin Absorber Measurements. For a sufficiently thin absorber, the transmission is determined by the average, over the resolution func- tion, of the cross section. Neglecting the variation of (E r /E) V2 , .E+RI2 &(E) = - I o{E) dE R Je—R/2 itan 2R 2(E - E r ) R r rj — tan 1 2(E - E r ) R\ r rl (116a) Curves of a/cr are plotted in Fig. 56 as a function of 2(2? — E r )/T, for a number of values of R/T. For R/T < 1, the main effect of the finite resolution is to depress the peak without greatly broadening the reso- nance (although the observed width at half-maximum is somewhat in- creased). Thus, for good resolution experiments, it is relatively easy to make the appropriate corrections to the transmission curves, and to determine the resonance parameters. (The work of McDaniel, Mil, on the 1.44 ev resonance of indium, see Fig. 7d, illustrates the method.) However, for R/T > 1 the observed width is essentially the resolu- tion. It is impossible, from such a measurement, to determine sepa- rately the resonance width or the peak cross section; a poor resolution thin absorber experiment can only yield the product a r, through Sec. 3D] Sources and Detectors 445 1 G w* i-4 / »-H / 7 5 4 / i— 1 ^ / / II *!•-. CN ^P o — II — 10 © »H «|t, II II CM i? fc) CD 1 ^^ K) cr -*3 W cj \ K) -"•o/a 5. 2 o « r 1 !> bXJ r> cm o Hi cl CJ a; IT e •n cd !> ^3 03 (H o CO Ph CD > ,0 O 01 "5 a br :a U +3 T cu (3 o 1*1 a <D «4-l K) fl O ci -^ (1 CD 01 n > J3 -^> a 3 71 TTn O .3 CO o till 1-1 3 O 03 o & «*-* CN -fJ Tl a. S «i id ^3 03 O 3 CO CO T) o 03 CD > II 02 CD u XI ^-s» O «=! CD ,£ <D Bl +» 1 eft -*> o bfl g 03 ^5 o bl) J3 C3 02 0) Ft O CD <D -*> 02 CD 1 CD a o > O T3 3 03 o O II W CO CD ft «© iO o $.2 (*«H 446 The Neutron [Pt. VII either the observed peak cross section or the area under the observed resonance: (a) The apparent (observed) peak cross section J is r _.B c{E r ) — <r — tan — R T R for - « 1 r -HOT] r /ir r\ r """r^-r) ""t >>1 (1I6a,) The reduction of the peak cross section (by the factor 1.571 T/R) makes it virtually impossible to use a thin absorber when R/T > 10. (b) It may be seen, by visual inspection of Fig. 56, that the decrease in the peak cross section is accompanied by a raising of the tail of the observed resonance, so that the area underneath the resonance curve appears relatively insensitive to the resolution. Actually (from Eqs. 116 and 116a) 2 fff(E) dE = fff(E) dE = - (T r (116b) independent of R ! Another possible effect of poor resolution is the failure to resolve close resonances. This difficulty is illustrated in Fig. 57, in which o/oq has been plotted, for two identical resonances separated by the energy D, for a number of values of R/T. The classic example of the phenomenon of close resonances is the case of iodine, which appeared, from early experiments, to have a single, anomalously broad resonance at ~40 ev. Careful investigation by Jones (J 12) established the existence of a num- ber of sharp resonances between 20 and 50 ev. Many of the relatively broad peaks in cross-section curves (G20, A2) observed above ~10 ev 1 Another resolution shape, for which the average can be performed analytically, is the Gaussian function of half-width R. In this case, HEr) = <r [1.665(r/fi)e o.693rV-R 2 erfc (0.833l7#)] S 1.476<r r/ft for R/T » 1 (116a") in which erfc (x) = (ir/4)* -JJe-^dt. 2 The integrals are conventionally and most easily performed between the limits — x and M. Since the entire contribution is for values relatively close to E r , this extension of the limits of integration introduces a negligible error. Sec. 3D] Sources and Detectors 447 in monochromator measurements on heavy nuclei are probably due to the failure to resolve such groups of resonances. (2) Thick Absorber Measurements. If a thick absorber is employed in a good resolution experiment, the transmission is ~0 over the main portion of the resonance; only the wings of the resonance can be studied in any detail. However, as we have seen above, thick absorbers are necessary in a poor resolution experiment to obtain transmissions which differ significantly from 1. The interpretation of the transmission curve for a thick absorber in a poor resolution monochromator experiment can be achieved by numeri- cal and graphical techniques, as. described by Havens and Rainwater (H45). The result of such interpretation — either by detailed fitting of the transmission curve or by integration of the absorption over energy — yields a value of the combination a T 2 . The interpretation requires a knowledge of the value of the resolution width, R, and of the shape of the resolution function; the results are, however, not very sensitive to the exact form of the resolution function. The fact that the combination a^T 2 determines the results of a poor resolution thick absorber transmission measurement can be seen by the following argument: The mean transmission at the energy E is I/I (E) = ( C R {E' ! E)e' NTn/n+i( - E '- IS ' )Vri] dE'\R(E) (117) where R(E',E) is the resolution function; the resolution width is de- fined as R(E) = fR(E',E) dE' (117') Assuming NT<r > 10, the integrand vanishes unless 4(£" — E T ) 2 /Y 2 ^> 1 ; hence, the integral may be replaced by I/h(E) ^ ( CR(E',E)e- NT " oTi/ ^ E '- Er) dE'\R(E) = F{E,E r ,a Y 2 ) (117a) The area under the absorption curve is /' (1 - ///„) dE = F'(E r ,a T 2 ) (117b) Since almost all monochromator measurements for E ^ 50 ev employ poor resolution and thick absorbers, a resonance in this region must be comparatively strong in order to be observed at all. There is no doubt that such measurements fail to detect a fair fraction of the resonances. 448 The Neutron [Pt. VII i i T-t II ^ J ii 1 ^ / <M o II ~* II *i ""St* ) 10 II \ ts I O OS r-i O to o o o Sec. 3D] Sources and Detectors 449 o 1 II "5 X > CM ii O 11 o=^ ^ I k -^ r \